The study relies on a differentiable 1D framework with a physical (Eq. ) and a biological component (Eq. ). The first right-hand term resolves the physics, that is the vertical diffusion of the tracer with a space-time-varying diffusion coefficient Kz. The second term SMS(C) accounts for the non conservative (Source Minus Sink: SMS) processes of tracer C, i.e. the BGC processes described in Eq. (). A no-flow boundary condition, defined as Kz∂C∂z=0, is imposed for the diffusion at each boundary. This Neumann condition guarantees no vertical outflows between the surface and the atmosphere nor between the water column and the seafloor. 1∂C∂t=∂∂zKz∂C∂z+SMS(C)

The NNPZD model (Eq. ) is inspired by . This model describes nitrogen concentration dynamics across five compartments: nitrate (NO₃), ammonium (NH₄), phytoplankton (P), zooplankton (Z), and detritus (D). The equations are defined over time and space (vertical dimension). However, to avoid excessive notation, the space dimension is omitted. They involve 16 fixed BGC parameters and 2 time- and space-varying physical forcings and state the time dynamics of these compartments as Ordinary Differential Equations (ODEs): 2dNO3dt=μNH4-GNO3κ+NO3e-ΨNH4P+λβ-NO3dNH4dt=-GNH4κ+NH4P-μNH4+ΓZ+φDdPdt=GNO3κ+NO3e-ΨNH4+NH4κ+NH4P-ΞP-ρ1-e-ΛPZdZdt=(1-γ)ρ1-e-ΛPZ-ΓZdDdt=γρ1-e-ΛPZ+ΞP-φD+∂Φsed(D)∂z

The 16 parameters (listed in Table ) represent physical and BGC sub-processes. The G factor symbolizes the photosynthesis process, denoted as G=ηαI(t,z)η2+α2I(t,z)2, with I(t,z)=I0(t)eξz+ζ∫0zP(t,z′)dz′ the photosynthetically available radiation (PAR) at depth z and time t, the PAR at the surface being denoted by I0. The total nitrogen uptake by phytoplankton is given by GNO3κ+NO3e-ΨNH4+NH4κ+NH4P. The zooplankton grazing upon phytoplankton is formulated by ρ(1-e-ΛP)Z, where a part of it goes to detritus through excretion and mortality with the term γρ(1-e-ΛP)Z. The terms μNH4, ΓZ, φD and ΞP stands for ammonia oxidation, zooplankton respiration, remineralisation and phytoplankton death processes, respectively.

The detritus compartment is affected by gravity and sinks with a sinking rate ω. This process is represented by ∂Φsed(D)∂z=∂ωD∂z for detritus, with a sedimentation flux Φsed defined through a third-order WENO scheme . This flux is equal to zero at the surface, given the absence of fluxes at the air-sea interface. In the absence of advection and horizontal diffusion, the sinking of detritus leads to the depletion of nitrogen from the system over time. Indeed, the sinking nitrogen remains too deep to be mixed to the surface during winter. In order to compensate for this loss, which is not compensated for by 1D dynamics, an additional term is introduced in order to replenish the nutrient profile and ensure a steady state . This so-called nudging , is formulated as λ(β-NO3). We assume that this nitrate supply occurs during the month of February, when the Mixed Layer is the deepest. We set the parameter λ to 0.05 d⁻¹ during February and to 0 d⁻¹ otherwise. As the depth of the mixed layer is among the key driver of nutrient fluxes from the deep ocean to the upper ocean , we parameterize reference concentration β with respect to the seasonally maximum of the mixed layer depth (MLD) and the NO₃ profile (Appendix ).

Following , we derive reference parameter values as detailed in Table . We tuned some parameters to ensure realistic nitrogen stock dynamics for the specified area for each compartments, especially a spring phytoplankton bloom, followed by a detritus peak and a subsequent zooplankton peak. Furthermore, it is preferable to avoid excessive P–Z oscillations with respect to prey-predator coupling.

Regarding the physical forcings in (Eq. ), we use the realistic North Atlantic subpolar gyre simulation spanning 7 years from 1 January 2002 and 31 December 2008 with a resolution of 2 km. This simulation is based on the CROCO model, which was developed from the Regional Ocean Modelling System (ROMS, ). We under sample from this simulation a set of 1-year time series of the solar irradiance I0 and of vertical ocean mixing profiles Kz, that we loop over 3 years in order to achieve a seasonal steady state. I0 has a 12 h time resolution and a 0.25° horizontal resolution, while Kz has a daily time resolution and a 40 km horizontal resolution. We focus on the upper ocean layer between sea surface and 350 m deep with 35 sigma vertical levels. Our study area is a 5°×5° box centred at the PAP (Porcupine Abyssal Plain) station in the North Atlantic ocean, where numerous studies are ongoing to better understand the carbon export processes in the meso-pelagic layer . Overall, the resulting 1D dataset is composed of 1-year time series {I0(t,p),∀t∈Ωt} and {Kz(t,p),∀t∈Ωt} defined in a time space Ωt at a horizontal location p∈Ωp.

For a given physical forcing time series, we run ocean BGC simulations according to model . For each simulation, we uniformly sample model parameters in a ±20% range of their reference value (see Table ). Each simulation includes a 2-year period of spin-up (w.r.t. Appendix ) with a constant nitrogen concentration as initial condition, fixed to β for NO₃ and 1% of β for the other compartments. As detailed in Sect. , β is defined according to the maximum annual mixed layer depth with respect to the mean vertical pattern of NO₃ concentration in the PAP area taken from the World Ocean Atlas 2023 . Overall, these ocean BGC simulations result in 3100 sets of 1-year daily sampled time series for the five BGC compartments.

Typically 3100 simulations (so called OSSEs) are available, of which card(Ωtrain)=2000 are used for training, card(Ωvalid)=1000 for validation, of the learning based scheme (presented in Sect. ). The remaining card(Ωtest)=100 datasets are reserved to test each calibration method.

We augment our dataset with noisy physical forcings to account for the uncertainties observed in ocean reanalysis dataset . It is widely acknowledged that state-of-the-art global ocean reanalyses cannot resolve upper ocean dynamics for horizontal scales below one hundred kilometres . Given the impact of fine-scale physical ocean dynamics onto biogeochemical ones , these uncertainties likely affect the calibration and reanalysis of ocean BGC dynamics using assimilation-based approaches. To simulate such uncertainties and biases in our dataset, we proceed as follows. Let us consider a given location at sea surface p and a time t and the associated physical forcings U(t,p)=(I0(t,p)Kz(t,p)). We define the noisy physical forcings U*(t,p) at space-time location (t,p) as: 3U*(t,p)=Ut,p+v(t) where v is a random time-evolving process stated as a first-order linear bivariate Gaussian auto-regressive process with standard deviation σv. The term v accounts for a spatial location uncertainty . The definition of the noisy forcings in (Eq. ) involves a spatial interpolation using a linear interpolation for both vertical mixing and solar irradiance. We consider three uncertainty levels: σv=0.15° (Level 1), σv=0.25° (Level 2) and σv=0.5° (Level 3). We point out that the considered spatial uncertainty is 95% of spatial shifts within ±2σv of the reference location. The interpolated profiles U(t,p) and U*(t,p) denote vectors that represent surface irradiance and vertical mixing values throughout the entire water column.

The resulting physical error patterns can be interpreted as misplaced mesoscale-to-submesoscale features seen in current physical reanalyses, depending on space-time range of the spatial shift (here, up to 0.5°). Due to the fine-scale variability in the input physical fields, especially for vertical ocean mixing profiles, they can include mean biases, amplitude shifts, changes in the vertical profiles. We acknowledge that this procedure cannot reproduce large-scale uncertainties and biases which may be observed in global reanalyses .

Besides physical forcing uncertainties, we also assess the impact of observation configurations onto calibration and reanalysis performance. We may distinguish two main categories of observing systems for ocean BGC processes: scientific cruises, BGC ARGO floats. These two categories differ in the associated space-time sampling.

Scientific studies sample specific areas and provide samples, as the Bermuda Atlantic Time-series Study (BATS) that uses conductivity, temperature, depth sensors (CTD, ) or bottles on rosette for specific analyses in order to detect specific temporal dynamics. The associated time sampling may typically range from days to months. The BGC ARGO network provides a global observation network, where each float delivers at a 10 d sampling rate vertical ocean profiles with a high vertical resolution, usually a few metres, of key variables, including among others temperature, salinity, oxygen, fluorescence, backscattering (a proxy for particle), and lately nitrate . Across these different observing systems, some compartments are easier to monitor than others. Especially, zooplankton and ammonium concentrations are notoriously difficult to estimate and require substantial installations to be deployed , ammonium being measured from water sampling and zooplankton from nets (or cameras for micro-zooplankton, ). By comparison, the assessment of nitrate, phytoplankton, and detritus concentrations is easier using proxies, such as chlorophyll-A for phytoplankton (directly measured from water fluorescence, ), or sensors directly measuring these parameters, such as ultraviolet sensors for nitrates . Overall, the resulting different observation configurations result in different space-time sampling patterns. Through the 1D ocean BGC setup considered setup, this study focuses on regular time sampling patterns from days to a month as well as the following four observation configurations for the water column:

Strategy 0 – Perfect scenario: As a baseline scenario, we assume the 5 compartments to be observed for the whole water column.

All observation configurations account for a zero-mean Gaussian additive measurement noise and a spherical covariance matrix with the following parameterization for the compartment-wise standard deviation, respectively 0.001, 0.0005, 0.04, 0.12, 0.08 (mmolm-3) for NO₃, NH₄, P, Z and D.

We introduce the following notations. Hereafter, index i refers to a time index according to a discrete set of I+1 time steps {t0,t1,…,ti,…,tI} such that ∀i,ti+1=ti+δt. Similarly, index j refers to an index along the ocean depth dimension according to a discrete set of vertical levels {z0,z1,…,zj,…,zJ}. Bolded variables, e.g. X, Y and U, refer to two-dimensional tensors according to time and depth, while a state at a specific time and depth is referred to as x(t,z). Especially, we denote Xi,j=x(ti,zj) that represents the value of tensor X at time ti and depth zj. In addition Xi=x(ti)=(Xi,0,…,Xi,J) refers to the vector of depth-related values of X at time ti.

Now, the calibration of a model consists in fitting observed fields to predicted ones by adjusting variables such as the model parameters or the initial state. In this case, the focus is on a BGC model, characterised by θ BGC parameters, which dynamic is represented by ODEs (see Eq. ). The operator Mθ(.) describes the dynamic of the system states from initial state conditions X0 and forcings U={U(t,z)}Ωt,Ωz with Ωt and Ωz the time and spatial spaces that are considered. Especially, the time space (resp. spatial space) can be subdivided into T time steps (resp. H spatial steps) equally separated by a fixed period δt (resp. a fixed distance δz), such that Ωt={t0,…,tT-1} and ti+1=ti+δt,∀i[[0,T[[ (resp. Ωz={z0,…,zH-1} and zj+1=zj+δz,∀j[[0,H[[). The observation of some ocean states is assumed with an observation operator H(.) that stands for the spatio-temporal sampling, and associated observation noise ϵobs. The variable Y denotes the observed states at different time steps. It is assumed that X and Y both have the same dimensions but Yi,j=0 if the state is not observed at time ti and depth zj. Typically, H is a diagonal matrix with a value of 0 for the non-observed states. The observation noise ϵobs features the various stochastic biases inherent to the observation processes. The system can thus be formulated as follows: 4x(t+1)=Mθ(x(t),u(t))y(t)=H(x(t))+ϵobs

The calibration problem can be regarded as an inverse problem. The objective is to design an operator that maps BGC parameters to the available information, i.e., Y, U* and θb (BGC parameters prior), to the actual BGC parameters θ. As mentioned in the introduction, particular attention is paid to the sparsity of the data, which can be formulated through H(.). Moreover, we note U*=U+νU the available forcing data, with U the actual forcings and νU={νU0,…,νUk-1} representing the spatio-temporal uncertainties. As generally assumed in assimilation or forecast problems, the considered forcings are U* since exact forcings U are not available . In this study, the problem at stake is a joint calibration that aims to calibrate the model parameters, correct the forcings and reconstruct the BGC dynamics. In particular, the objective is to get an operator, denoted by K(.), s.t. θ^,X^,U^=K(θb,Y,U*). The notations θ^, X^ and U^ respectively refer to estimated BGC parameters, reconstructed states and corrected forcings.

The complexity of a BGC model directly correlates with the difficulty of constraining it. The correlation of its parameters and the redundancy of the observations can result in certain parameters compensating for each other, leading to erroneous values: the system is under determined . A preliminary sensitivity test has been conducted in order to consider only sensitive parameters in the calibration with observed BGC state concentration. In other words, the calibration is based on only 8 out of the 16 BGC parameters, with the remaining 8 being assumed to be known (see Appendix ).

Among the state-of-the-art data assimilation (DA) methods, this study proposes the variational data assimilation scheme as a reference calibration method (; ; ; ; ). This weakly constrained 4Dvar scheme seeks to identify an optimal set of BGC parameters θ and initial states Xi, that minimises the error of BGC states simulation Mθ(Xi,Ui), based on a set of observation data Y (Eq. ). We consider ocean states at times {t0,…,tT-1} over a period Tδt. In addition, the period Tδt can be divided into d sub-periods of Δ time steps, such that dΔδt=Tδt (see Fig. ). The weakly constrained nature of the 4Dvar scheme accounts for a dynamic error. This error is handled by dividing the studied time series into sub-windows (here we consider sub-windows of Δ time steps) and by ensuring the model accuracy over these short simulated periods. For this, we denote X0d-1=(X0,XΔ,⋯,X(d-1)Δ) as the state at the first time step of the d Δδt-time windows. In the same way, X1d-1=(X1,XΔ+1,⋯,X(d-1)Δ+1) refers to the second time step of the d Δδt-time windows. 5J(X,Y,U)=Jobs(X,Y,U)+Jmodel(X,U)

The primary right-hand side term of the variational cost is a fundamental component in variational methods, ensuring that model simulation aligns with observed states (defined in Eq. () and illustrated as Err_obs in Fig. ). Each (Δ-1)-time step of each Δδt-time window is simulated and compared to the observed state when it is available. We denote by Mθ(i)(X0,U0:i-1)=MθMθ(i-1)(X0,U0:i-2),Ui-1 the application of the dynamical model operator M(⋅) defined in Eq. () between step i-1 and step i. This notation facilitates the writing of simulated states over several time steps and leads to Mθ(1)(X0,U0)=Mθ(X0,U0)=X1. The simulation of BGC states X from time steps 0 to i, requires the initial state X0 and forcings at each time step U0:i-1. Thus, Xi=Mθ(Xi-1,Ui-1) can be written as Xi=Mθ(i-1)(X0,U0:i-1). Moreover we introduce the notation: 6MθΔ-1Xi,Ui:i+Δ-1=(Mθ(0)Xi,Ui,Mθ(1)Xi,Ui:i+1,⋯,Mθ(Δ-1)Xi,Ui:i+Δ-1) that returns a matrix with the simulated states at each step from step 0 to step Δ. In combination with X0d-1 (defined in Sect. ) that gathers the state at the first time step of every Δδt-time period, the term MθΔ-1(X0d-1,U0:Δ-1d-1) indicates the Δ simulated terms of each of the dΔδt-time periods. The notation ||Y||R-12 product stands for YTR-1Y, with R a covariance matrix relative to the observation noise. In our case R-1 is a diagonal matrix with [1/σ2] as coefficients, σ the state error standard deviation. 7Jobs(X,Y,U)=HMθΔ-1(X0d-1,U0:Δ-1d-1)-Y0:dΔ-1R-12

The second right-hand side term, defined in Eq. (), aims to constrain the time consistency of the BGC states for the successive time windows. Following an approach similar to a weak-constrained 4DVar scheme, this measures the model error between a state X(k+1)Δ and the BGC states simulated from the Δ previous time steps, i.e., Mθ(Δ)XkΔ,UkΔ:(k+1)Δ. 8Jmodel(X,Y,U)=Mθ(Δ-1)X0d-1,U0:Δ-1d-1-XΔd-1B-12 with B the model error covariance. We consider a diagonal covariance matrix whose diagonal terms BC,C=τDAσC2, where σC2 denotes the variance associated with BGC state C. The coefficient τDA is a positive scalar that controls the relative weight of the observation and model costs in Eq. (). Based on cross-validation experiments, we set τDA=10-3 for all our experiments.

Numerically speaking, it is assumed to have the adjoint operator or a differentiable implementation of the model M w.r.t. parameters and states. Then we solve the above minimization with a fixed-step gradient descent, beginning with a first guess of the BGC parameters and the initial states as illustrated Fig. . The minimization process typically involves 1000 gradient descent steps, implemented using an Adam optimizer. The assimilation process entails optimising the BGC initial conditions, which are not directly observable. In this case, we use as a first guess a constant matrix with the mean value of the state over the time and spatial spaces, i.e., X=1E[Y] the averaged observed states, with 1 a matrix full of 1. The BGC parameter first guess θb is a draw following the uniform distribution between 0.8 and 1.2 times the reference value of the parameter (cf. Table ), s.t. θb∼U(0.8×θref,1.2×θref). Therefore, the control variables in this study are the BGC parameters θ and the initial state estimate X0d-1 at the beginning of the d sub-periods. As commonly assumed in operational systems for biogeochemistry, forcings are assumed to be error-free, i.e., U=UTrue. In other words, we assume the forcing uncertainty to be part of the model error covariance in the minimization process.

Among the variational methods, ensemble variational methods enable to consider various members of a distribution simultaneously. Here, we consider an ensemble of L realizations of noisy forcings U(l)*=U‾+νU(l) for each sample, with l∈[[1,L[[ the number of the member and U‾ the mean forcings. Assuming the forcing uncertainty distribution is centred on 0 , i.e., 1L∑lνU(l)=0, the noisy forcings U* is non-biased, and 1L∑lU(l)*=1L∑l(U‾+νU(l))=U‾. Especially, one could consider solving the calibration problem for this estimated mean forcing, i.e. θ^,X^=∑lKDA(θb,Y,U(l)*)=KDA(θb,Y,U‾).

In this paper, we propose a neural-based scheme as an alternative to DA. The method delineates the calibration problem as the supervised learning of a neural mapping between inputs, i.e., observations and forcings, and model parameters variables. As illustrated in Fig. , we benefit from the versatility of deep learning schemes to jointly address the calibration problem, the denoising of the forcings as well as the reconstruction of the BGC dynamics. Formally, the neural network acts as an operator (θ^,U^,X^)=KNN(θb,Y,U*) with Υ as the parameters of the neural network using the notations introduced in Sect. .

For the training of neural operator KNN, we assume a representative dataset gathering error-prone and error-free forcings, BGC parameters, true BGC dynamics and associated observation data. The training phase amounts to minimizing the following loss with respect to neural network parameters Υ: 9LOSS=∑a(θ^-θ)2+b(U^-U)2+c(X^-X)2 with a, b, and c weighing factors for the three MSE terms, namely the BGC model parameters, the forcings and the BGC states. We tune the value of these parameters empirically through cross-validation experiments according to a=1, b=5, and c=5. Our training configuration exploits a stochastic gradient descent method using an Adam optimizer with a learning rate of 1×10-3 and mini-batches of size 256.

Regarding the neural architecture, we consider a state-of-the-art UNet . This multi-scale architecture is widely used in imaging applications including for ocean studies . The computational graph of the architecture is illustrated in Fig. . More complex architectures, combining for instance UNet with residual and attention mechanisms , could be considered. We favoured the trade-off between computational efficiency and computational complexity in our experiments. The model is composed of a central block (ensemble of layers) with a contracting (encoding) path for the capture of relevant features of the input and a symmetric expanding (decoding) path that enables precise localization of these features. We complement the classic UNet architecture with three task-specific blocks which address the estimation of BGC parameters, BGC states and physical forcings. We use a channel-wide normalization of the inputs and outputs according the mean and standard deviation of each variable. This ensures effective training of the model and equitable assessment of the estimated variables. The preprocessing also includes the space-time interpolation and concatenation of the observed state and the physical forcings. The input results in a tensor of dimension (Nbatch×(Nch×(Nz+1))×2NT), where Nbatch is the number of samples per batch. Nch=7 is the number of channels, i.e., the number of state compartments (5 for NO₃, NH₄, P, Z, D) and the number of forcings (2 for Kz and I0). Nz=34 is the number of layers at which tracers are resolved (vertical diffusion is computed at the interfaces, i.e., at Nz+1=35 depths), and NT is the time-window length in days (since the forcings are provided at 12 h intervals).

The learning-based method, described above, simultaneously estimates the parameters, the initial BGC conditions, and the physical forcings. As shown in preliminary experiments (see Appendix ), the DA-based method produced more accurate parameter estimates than the UNet in error-free physical configurations. These findings suggest the potential benefit of an hybrid approach, as sketched in Fig. , combining both the UNet and 4Dvar frameworks. This hybrid scheme operates in two stages, starting with the UNet (presented in Sect. ). Its outputs–namely the BGC parameters, the BGC state variables, and the physical forcing–are then passed as inputs to the 4Dvar method. The parameters and BGC state estimated by the UNet serve as a first guess for the assimilation process, while the UNet-derived physical fields are used as forcings. We expect the subsequent data assimilation step to refine model parameters and the BGC state, owing to the expected smaller errors in the physical forcings and the use of first guesses that are closer to the true state.

In this section, we describe our evaluation procedure for both the targeted model calibration problem as well as the reconstruction of the physical forcings and of the ocean BGC dynamics. Specifically, we detail how each scenario is considered and the metrics used to compare the three schemes: the 4Dvar DA scheme, the UNet scheme and a hybridization of the two methods.

Two categories of metrics are considered: those computed in the space of BGC model parameters, and those defined from BGC state dynamics. The first set of metrics involves statistics (e.g., mean, standard deviation…) and normalized error values for each BGC model parameter, expressed as θ-θ^θ‾, where θ‾ is the mean value of the parameter (given in Table ), θ is the true parameter value, and θ^ is the estimated value. For the second category of metrics, we proceed as follows. Given a predefined dataset of error-free forcings and initial conditions, we simulate a pair of BGC state time series according to Eq. () using the estimated and true BGC model parameters and focus on depth-integrated quantities to assess the difference between any such pair. Let us denote by Sc(t)=∫Ωzc(t,z)dz for the depth-integrated stock of a tracer of concentration c at time t (see Fig. f).

Given the stock Sc(t) for the true BGC model parameters and the stock Sc^(t) for the estimated ones, we define the following three metrics:

The correlation, computed as corrSc(t),Sc^(t), evaluates the pattern of the estimated stock.

We report in Fig.  the evaluation of the two calibration schemes presented in the Sect.  and  for a reference scenario. This scenario corresponds to the lowest uncertainty level (Level-1 uncertainty scenario, i.e. 0.3° of spatial uncertainty in the physical forcings as described in Sect. ) and the CTD + Rosette observation strategy (Strategy 1 cf. Section ) with a 10 d sampling rate. This reference scenario is rather optimistic compared to the characteristics of operational systems .

Figure a–c shows the distribution of the correlation, shift and amplitude ratio metrics between the nitrogen stock obtained from the calibrated BGC parameters and the actual BGC parameters, for 100 different 1D simulations. The correlation score results in a clear shift towards lower values for the DA-based scheme with a mean correlation score of 0.88 and a minimum value of -0.49 compared to a mean value of 0.99 and a minimum at 0.68 for the UNet. We observe a consistency of the two approaches with better correlation values for NH₄ and Z compared to NO₃, P and D. As depicted in Fig. b, the resulting stocks in the NH₄, P, Z and D compartments are commonly shifted, up to 10 d for the UNet and up to 28 d for the 4Dvar method. Figure c illustrates that the predicted stocks are more often under-estimated or over-estimated when using the parameters from the DA-based scheme. This is supported by a larger standard deviation of the amplitude ratio for the DA-based scheme compared to the learning-based approach (0.28 vs. 0.08).

Figure d presents the normalized error of the estimated parameters (defined in Sect. ). While the DA-based approach leads to parameter estimate error between -0.38 and 0.40, the UNet reaches lower error values between -0.20 and 0.18. On average, it reduces the estimation error standard deviation by a factor of 2.8. As expected, some BGC parameters such as ρ, Γ and β show a lower estimation uncertainty compared with the other parameters consistently for the two calibration schemes. This likely relates to differences in the identifiability of BGC model parameters. Unsurprisingly, parameters γ, Ξ and ω, which drive mortality (of P and Z) and particle sinking processes, are known to be under-constrained in classic model-based frameworks . Importantly, the learning-based scheme depicts much lower differences of the calibration uncertainties across model parameters.

Based on the quantitative evaluation presented previously, in the subsequent sections, we focus on the learning-based calibration scheme to study the impact of physical forcing uncertainties and observation configurations into the calibration performance.

Here, we assess the impact of physical forcing uncertainties onto the performance of the ocean BGC model calibration using the learning-based approach. As discussed in Sect. , a sensitivity analysis is conducted for three distinct scenarios, designated as case-1, case-2 and case-3 corresponding respectively to mean horizontal uncertainties of 0.3, 0.5 and 1° in (Eq. ). The observation configuration involves a 10 d sampling of the full BGC state over a 120 d period from February to June. The evaluation dataset comprises 100 samples, each sample being associated with an ensemble of 10 noisy forcings. For each sample, we consider the ensemble mean as the estimated BGC parameters for both the DA-based and learning-based schemes. We report in Fig.  the synthesis of these experiments.

Figure  reports the calibration performance of the learning-based scheme for the three forcing uncertainty scenarii introduced in Sect. . Overall, the performance does not show a strong dependence on the uncertainty level. The averaged correlation scores ranges between 0.98 and 0.99 with larger values for NH₄ stocks. The shift scores have close standard deviations with larger lags for P and D stocks. Finally, all the amplitude scores feature the same distribution whatever the considered scenario or state. With regard to the BGC parameter errors illustrated in Fig. d, similar patterns are observed. The standard deviation of parameters varies by less than 0.05 for all forcing uncertainty levels. Parameters α, γ and φ however demonstrate a heightened degree of sensitivity. The standard deviation values range between 0.01 and 0.017 in the three scenarii under consideration. For the remaining parameters, the standard deviation does not vary by more than 0.007 between the scenarios.

As described in Sect. , we carry out experiments to evaluate the sensitivity of the calibration performance with respect to the observation configurations, using the learning-based approach. As depicted Fig. , we first focus on the impact of the time sampling rate assuming a CTD + Rosette observation configuration for the vertical profiles. We recall that the CTD + Rosette configuration involves observations of NO₃, P and D concentrations at all vertical levels and of NH₄ and Z concentrations at 5, 25, 50, 100, 150, 200 and 350 metres deep. The four scenarios differ in terms of the time sampling rate: 60 d (brown distributions in Fig. ), 30 d (red distributions in Fig. ), 10 d (purple distributions in Fig. ) and 1 d (blue distribution in Fig. ). As illustrated in Fig. a–c, the correlation, shift and amplitude ratio metrics demonstrate a marked difference for sampling rates of 1 and 10 d vs. 30 d and 60 d. For instance, we observe high values of the correlation score (resp. 0.994 and 0.987) for 1 and 10 d sampling rates, compared to 0.97 and 0.957 for 30 and 60 d sampling rates. Similarly, the average of the shift score remains below 0.66 d for a sampling up to 10 d and is above 1.10 from a 30 d sampling. As the time sampling increases from 1 to 60 d, the amplitude ratio standard deviation concomitantly rises, reaching 0.05, 0.08, 0.13 and 0.14, respectively. Regarding BGC parameter error in Fig. d, the standard deviation increases with the sampling period, with values of 0.03, 0.05, 0.09 and 0.10 recorded for time sampling rates from 1 to 60 d. Some parameter depicts a greater sensitivity to the observation sampling rate. For instance, parameter Γ exhibits a steep increase to 0.1 of the standard deviation for the 60 d scenario, whereas for 1, 10 and 30 d, the standard deviation remains between 0.02 and 0.06. It is noteworthy to mention that parameters Ξ, φ and ω have the same error distribution whether the states are observed every 30 d or every 60 d. As illustrated in Fig. , the simulated BGC trajectories typically involve a one-peak pattern with a duration of a few weeks for each tracer. Therefore, 30 and 60 d sampling rates likely poorly inform the duration and the amplitude of the peak. As an illustration, we report in Fig.  a scatterplot of the NMSE of BGC parameters vs. the NMSE of the reconstruction of the BGC state from an interpolation of the observations. The 30 and 60 d sampling configurations clearly result in both high estimation errors for BGC parameters and of the reconstruction error, the latter being regarded as a proxy of the quality of the monitoring of the peak-shaped pattern of the BGC states.

Besides the time sampling rate, we also assess the impact of the sampling pattern along the depth dimension according to the following four strategies: an ideal scenario in which each state is sampled at all depths (light blue distribution), the CTD sampling strategy (red distribution), the Rosette sampling strategy (brown distribution) and the CTD + Rosette sampling strategy (purple distribution). All of the aforementioned scenarios are described in Sect. . All experiments exploit here a 10 d sampling rate. The first scenario serves as a baseline to assess an upper bound of the calibration performance. As illustrated in Fig. , there is a clear lowering of the calibration performance when considering the CTD-only and Rosette-only strategies compared to a “All obs” observation configuration. Among the different BGC states, we observe the largest impact for NO₃ and P in Fig. a–c. Figure a–c underline lower reconstructed states correlation, especially for NO₃ with a score value of 0.98, as compared to 0.99 for the 'all obs' configuration. It is noteworthy that the standard deviation is four to five times higher in the latter case. It is also observed that there is a marginal increase in delay for every reconstructed stock except NO₃. Shifts are 0.3 to 0.5 d higher for CTD-only and Rosette-only configurations in comparison with a “all obs” configuration. Regarding BGC parameter errors, Fig. d strengthens the discrepancy between the CTD-only and Rosette-only observation configuration for ρ, γ, Γ and φ when compared to an idealised scenario. The former scenarii exhibit a range of errors that is 1.4 to 3.5 times higher than the “all obs” scenario, and a range of standard deviation values that is 1.5 to 3.9 times higher.

We also evaluate the potential impact of the proposed calibration schemes onto the reconstruction of the 1D BGC dynamics when considering forcing uncertainties and a realistic observation configurations. We focus on the baseline scenario used in Sect.  to benchmark the DA-based and learning-based calibration schemes (we also report a complementary experiment involving unobserved components is presented in Appendix ). As described in Sect.  and , both approaches considered lead to the estimation of the time series of the BGC states. Furthermore, learning-based schemes also yield a corrected version of the forcings. This suggests the consideration of the third hybrid approach, presented in Sect. . It should be noted that, unlike the experiment described in Sect. , the BGC parameters are not treated as control variables in the DA scheme. This choice is based on a preliminary experiment (reported in Appendix ), which shows that the hybrid method cannot reliably optimize both the BGC parameters and the BGC state variables simultaneously, due to a low robustness to physical forcings, which are not corrected with sufficient accuracy by Unet. Consequently, the hybrid scheme focuses on refining the BGC state trajectories.

Table  synthetizes the results of these experiments. The performance of the reconstructions is assessed through the NMSE of the five BGC variables. For the three approaches that have been benchmarked, ensemble-mean estimates are derived using the reconstructions issued from each of the ten forcing members (see Sect. ). Similarly to the calibration experiments, the findings indicate a substantial impact of forcing uncertainties on the performance of the DA-based approach, with two to three times higher NMSE values observed in comparison to the learning-based scheme (e.g., 0.012 vs. 0.007 and 0.241 vs. 0.084 for NO₃ and P). It is noteworthy that the hybrid approach enhances the reconstruction performance for all BGC variables with the exception of NO₃. For instance, the NMSE is divided by two for P and Z concentrations (respectively 0.085 vs. 0.047 and 0.017 vs. 0.008). It is also important to note that this reconstruction performance significantly improves a direct interpolation of the observation data (e.g., 0.029 vs. 0.769 in average). Regarding the standard deviation, presented in Table  and computed from the 10-member ensembles for each sample, the learning-based scheme provides the lowest uncertainty for P, Z and D concentrations and the UNet based scheme provides the lowest uncertainty for reconstructed NO₃ and NH₄ concentrations. The hybrid method provides the highest standard deviation, i.e. a larger distribution within the 10-member ensembles. According to the distribution representativeness, the hybrid scheme yields the distribution that least underestimates physical uncertainties, resulting in a more realistic overall distribution. The former method results in a distribution that comprises 80.6 % of the true value, in comparison to 42.1 % and 65.8 % for the DA-based and the learning-based schemes, respectively.

Figure  illustrates one sample (i.e. one 10-member ensemble) among the 100 presented in Table , representative of the average performance of the currently presented experiment (other samples are available in Appendix ). The following comparison is made of the reconstructed BGC stocks using a 4Dvar-only-based (blue), a UNet-only-based (orange) and a hybrid (green) scheme on one sample, and their relative uncertainty space is denoted as the space between the lower limit of the μ-2σ and the μ+2σ. As demonstrated by the latter figure, the reconstruction performance of each state is satisfactory for every reconstruction scheme, provided that the initial guess is a noisy 10 d linear interpolation of the observed states (red dots). Especially, the methods provide a relatively accurate representation of the bloom event, which occurs between April and May. Furthermore, it displays the uncertainty time evolution of the different reconstruction schemes. The uncertainty related to the DA-based schemes tends to be higher during the bloom period, with some bulging shapes, that is attributable to the 10 d division of the reconstructed period. This uncertainty remains constant (approximately ±30 mmolm-2) throughout the entire reconstructed period with the UNet-based scheme. The hybrid method also yields uncertainty estimates that combine characteristics of the last two methods: they are larger during the bloom period, similar to the DA-based reconstruction scheme, while remaining as smooth as the uncertainties obtained with the UNet-reconstructed stocks. Overall, we obtain higher uncertainty levels that more effectively encompass the true state values.

The proposed solution supports the relevance of learning-based schemes to enhance biogeochemical simulation in spite of spurious physical forcings. The calibration of ocean BGC models is among the major sources of uncertainties in simulating, forecasting and reanalysis ocean BGC dynamics . State-of-the-art systems generally rely on data assimilation approaches and expert knowledge . This approach achieves better calibration performance relative to a state-of-the-art learning-based technique when physics is perfectly known, as demonstrated for a 0D framework in and here for a 1D framework (see Appendix ). Nevertheless, these results remain valid in an idealised, error-free setting. In realistic applications, the method may become computationally expensive and highly sensitive to uncertainties in ocean physics and limited observation configurations. This has prompted researchers to investigate alternative approaches. The exploitation of learning-based methods has gained significant interest in ocean science. They have shown promise in uncovering key processes from noisy data as well as tackling inverse problems such as model calibration problems . This study contributes to this ongoing research and explores end-to-end learning schemes to enhance the representation of ocean BGC processes. We extend our previous work to realistic 1D ocean BGC OSSE. The experimental setup provides an intermediate-complexity testbed. This enables a more precise evaluation of how uncertainties in ocean physics and the sampling patterns of observing systems influence the calibration of ocean BGC models.

This approach achieves superior calibration performance relative to a state-of-the-art learning-based technique when the physics are perfectly known, as demonstrated in experiments conducted in a 0D framework and a 1D framework (see Appendix ). Nevertheless, these results remain valid in an idealised, error-free setting. In realistic applications, the method may become computationally expensive and highly sensitive to uncertainties in ocean physics and limited observation configurations.

Our experiments highlight a clear impact of forcing uncertainties on model calibration performance for a variational data assimilation scheme. Here, forcings' uncertainties as presented in Eq. () do not account for possible large-scale uncertainties and focus on small-scale processes, spanning horizontal scales from 1 to several tens of kilometres. This likely underestimates the uncertainties in real ocean physics reanalyses. At sea surface, current reanalyses typically resolve dynamics down to horizontal scales of about 1–2°, given the present configuration of satellite-derived observations . As pointed out in previous studies , such uncertainty levels strongly degrade the calibration performance of variational data assimilation schemes, which implicitly assumes error-free forcings. In the worst-case scenarios, the resulting calibration can substantially distort estimates of BGC parameters and simulated BGC patterns. To deal with forcing errors, the model-based calibration scheme can introduce substantial biases into BGC parameter estimates. There are noticeable shifts of the seasonal blooms up to a month for phytoplankton and peak amplitudes for nutrients, zooplankton and phytoplankton can be misestimated by factors exceeding 1.5, even under low-uncertainty conditions (see Fig. ). By contrast, the learning-based scheme reaches a greater robustness for all tested configurations. The improvement stems from the end-to-end learning strategy. The neural network is trained to handle both forcing uncertainties and sparse observation configurations using a training dataset of around 3000 samples. This supervised learning paradigm cannot be implemented with this classic variational DA scheme, given the offline resolution of the physical field. The latter setting resembles an unsupervised approach, where the variational cost (Eq. ) acts as the training loss. These findings provide evidence that supervised learning paradigms can outperform unsupervised methods for model calibration problems. Future work could explore benchmarks between coupled data assimilation methods and learning-based approaches for ocean BGC model calibration. Since the computational cost of such studies is currently prohibitive for state-of-the-art Earth system models, the design of intermediate-complexity alternatives would be valuable (e.g., 1D or multi-layer-2D settings).

This study also highlights the impact of in situ sampling patterns on both calibration and reanalysis tasks. Several studies have documented the influence of data availability on BGC model calibration. Regarding the depth-wise sampling pattern the Rosette-like sampling configuration yields results comparable to those obtained from full-depth observations of the five modelled states as shown in Fig. . However, the CTD-like sampling scheme reveals a critical role for ammonium and zooplankton information in constraining BGC parameters. This result is consistent with earlier studies , which emphasise the value of zooplankton data in BGC model calibration. The temporal frequency of the sampling is also a key consideration, as deploying specialised equipment such as Rosettes to collect specific BGC components is costly. Our results underscore a marked difference in calibration performance between 10 d and 30 d sampling intervals, with the latter producing poor estimates of BGC parameters. Consistent with the system's temporal variability and inertia – i.e., its characteristic seasonal bloom – the 10 d optimum agrees with the weekly sampling interval identified as sufficient by . Further studies could investigate the added benefit of including complementary observations, such as temperature, salinity, dissolved oxygen and pH, in model calibration . This involves increasing the complexity of the model, e.g. by incorporating the temperature dependence of parameters and integrating carbon components .

While we point out the possible combination of the proposed neural schemes with a variational data assimilation to enhance the reconstruction of ocean BGC dynamics, the development of hybrid approaches jointly exploiting model-based and learning-based paradigms is an active research avenue in data assimilation. They can help addressing DA challenges such as adjoint model computation, model parameterisation, model error correction and computationally-efficient optimizers . Our future work will explore how such hybrid schemes could improve furthers the calibration, simulation and reconstruction of ocean BGC dynamics.

The ability of the proposed learning-based scheme to deal with physical forcing uncertainties is limited by how these uncertainties are accounted for in the training dataset. The principle of neural-based methods is to learn a mapping from an input space to an output space. Such methods are effective within the range of input-output relationships exhibited in the training data. Their performance is thereby constrained by this data and this generalisation performance beyond the distribution of the training dataset is a challenge . This aspect requires the training data to closely represent the conditions expected at inference time. In our study, input-data interpolation has proven effective in accommodating different spatial and temporal observation strategies. However, the neural network's performance depends on the level of forcing uncertainties in the specific scenario as described in Sect. . The input-output relationships learned by the model then differ between models trained with level-1, level-2 and level-3 forcing uncertainties. This suggests that maintaining the same level of data quality is essential to ensure reliable performance between from training to inference. As physics products can be resolved at different horizontal scales with different model parametrisations , the nature of any associated errors may also be subject to variation. It would be worth investigating whether a single model can accommodate different types of forcing errors to both extract valid signals and correct erroneous components from the BGC observed states. As presented in , multi-mode networks effectively capture different modes in input data, such as region dependent processes. Similarly, several learning-based models trained on different types of physical errors could be combined.

DA-based methods can straightforwardly accommodate different observation datasets with prescribed noise levels. By contrast, the present learning-based approach involves training a neural scheme for each observation configuration – namely, specific sampling patterns and observation noise parameters – as shown in Figs. –. For regional case studies, this assumption of constant noise is reasonably justified . Scaling this approach to larger spatial domains, however, naturally introduces spatial variability in BGC distributions as well as time-varying observation errors. In learning-based frameworks, it is standard practice to train on large ensembles of samples spanning a wide range of input distributions, thereby enhancing robustness to diverse density regimes . The versatility of neural schemes also enables to consider additional input variables to account among others for the observation configuration (e.g., noise level, sampling patterns, observation types), uncertainty levels of forcing or ocean provinces . This may result in an increase of the complexity of the considered neural architectures. But, one would expect such all-purpose architectures to reach similar performance that configuration-specific ones with a significantly greater generalization potential beyond the training datasets.

It is important to note that BGC model errors, which we do not account for in our OSSEs, could significantly affect the application on real observations. The neural network here is trained to retrieve BGC parameters for known equations that fully reproduce the observed states, apart from observation noise. The underlying model assumes that BGC model errors arise primarily from incorrect calibration of the parameters. Nonetheless, simulated BGC state errors can also result from erroneous BGC process formulations or from coarsely resolved scales . Several studies have explored learning-based tools as a solution to predict unresolved processes . The ability to separate modelled from unresolved processes, when combined to model calibration, offers a new perspective to account for model error more effectively. This is of particular relevance given the increasing complexity of the BGC models and their computational cost. Such methods could enable calibration of both simple BGC models, such as the one described in Sect. , and more complex models as in , by calibrating the relevant processes and correcting the unresolved components.

In this study, we assume that BGC states are directly observable. In real world experiments, nitrate and ammonium concentrations are measurable using sensors or bottle sampling . However, phytoplankton, zooplankton and detritus are much harder to directly observe. Phytoplankton is typically inferred from chlorophyll-a and fluorescence measurements . demonstrate that BGC model calibration is influenced by the choice of phytoplankton proxy. Additionally, in-situ instruments such as Underwater Vision Profilers, nets and backscattering provide information within specific size ranges . However, these measurements represent only a fraction of actual zooplankton and detritus concentrations . More complex models can explicitly represent chlorophyll and partition zooplankton and detritus into size classes . This approach mitigates proxy biases, but at the cost of increased model complexity. Bridging such schemes and the proposed learning-based framework is a challenge for future work.

Our study supports the relevance of the proposed learning-based schemes for the calibration and reanalysis of ocean BGC dynamics through numerical experiments for an intermediate-complexity 1D NNPZD case-study. These results naturally raise questions on the applicability of these approaches to real ocean BGC dynamics. As discussed in the following, we distinguish three main scientific challenges to scale up to real-world case studies: namely the generation of training datasets for real-world case-studies, accounting for real observing systems and dealing with the full range of space-time variabilities of ocean dynamics on a global scale.

Training datasets play a pivotal role in the performance of learning-based schemes . State-of-the-art neural schemes may require anywhere from thousands to billions of training samples, depending on the complexity of the task and the neural architecture . In line with image-to-image and signal-to-signal mapping approaches , this study demonstrates that relevant calibration and reanalysis neural schemes can be trained using a few thousand samples, each corresponding to a one-year time series. Importantly, the training dataset spans the full range of ocean BGC parameters, forcing conditions, associated uncertainties, and observation patterns. To our knowledge, no such simulation dataset is currently available for state-of-the-art 3D ocean BGC models. However, while the associated computational complexity remains affordable in the present 1D case study, it could become prohibitive when scaled up to state-of-the-art 3D ocean BGC models. For illustration, a one-year simulation of a 1/4° NEMO-PISCES configuration typically requires between 1200 and 2000 hCPU . Therefore, the generation of a training dataset of few thousands of training samples similar to that considered in the current study would represent a computational effort similar to running ensemble simulation over decades with an ensemble size of 100 members. The development of neural emulators of ocean models, which emerge for ocean physics, will likely provide a solution to address this computational bottleneck. To date, the development of emulators has been primarily focused on the physical components of the ocean. propose realistic representation of the global ocean circulation. These data-driven models represent a novel alternative to conventional ocean physical models. For ocean BGC dynamics, current demonstrations focus on specific integrated or sea surface variables, such as sea surface chlorophyll . Neural emulators for 3D ocean BGC are likely to emerge rapidly. Combined with ocean physics and atmosphere emulators, they would provide new means to transfer our study to realistic ocean BGC frameworks, both in terms of training datasets as well as in terms of fully-controlled benchmarks.

A second challenge that has been identified in the application of the framework to real-world case studies is the ability to exploit actual observational datasets, as pointed out in the previous section. While the proposed approach readily extends to known observation models, i.e., when the relationship between the observations and the ocean BGC states is explicitly defined, the application to observations, which do not directly relate to a BGC state variable, will require additional methodological developments. We may cite among others ocean colour products and sea surface chlorophyll , sediment traps and isotopic tracer measurements . In this context, it could be beneficial to employ neural schemes trained to relate measured quantities and modelled variables, as explored in the processing of Argo float data . Such neural schemes could also be integrated into the simulation process used to generate training datasets, thereby enabling the training of end-to-end neural schemes that map directly from the observation space to the model parameter space. In such methodologies, the ability to represent observational and modelling uncertainties will be key for applications to real observation datasets.

Another key challenge in ocean dynamics lies in accounting for the full spectrum of spatio-temporal variability, including those driven by climate change. In this study, we focus on a specific region of the North Atlantic. The literature recognises distinct ocean bioregions, each characterised by specific interactions between physical and biogeochemical processes . Existing bioregion classifications could inform the development of region-specific data-driven models , similar to the approach adopted here; meanwhile, recent advances in deep learning demonstrate that neural architectures can scale effectively with task complexity . For instance, our experiments employ neural architectures with approximately 130 000 parameters. By contrast, state-of-the-art neural schemes in weather forecasting and computer vision typically comprise tens of millions to billions of parameters . Mechanisms – such as attention blocks – provide robust means of modelling complex, context-dependent relationships, thereby enabling models to adapt to intricate dynamics. However, scaling up to more complex neural architectures is likely to require larger training datasets, underscoring the interconnected nature of the challenges discussed here.