With the explosion of big data and artificial intelligence (AI), research on innovative approaches that leverage the power of AI for flood forecasting and hydrological modeling has demonstrated significant efficiency and advantages in terms of accuracy and computational cost compared to traditional rainfall–runoff models . Hydrological modeling approaches have evolved from early rational methods that use simple linear equations relating peak discharge to rainfall intensity, through unit hydrograph theory, to complex statistical models that rely on empirical relationships derived from observed data. In parallel, physically-based modeling approaches emerged based on the blueprint model , which provides a complete theoretical structure for watershed processes. However, the descriptive equations used for each process require significant simplifying assumptions in real-world applications . In the pursuit of robust runoff simulation and/or of physical realism, the “resolution–complexity continuum” has been explored over the past five decades with approaches of varied complexity, from black-box to fully physics-based. Conceptual models offer a middle ground, representing watershed processes through simplified conceptualizations that incorporate physical principles. While conceptual models provide more practical implementations than physically-based models and retain more physical interpretability than empirical approaches, they often suffer from several limitations: lack of scale-relevant laws, intrinsic uncertainties in their structure and parameterization, no explicit link between hydrological parameters and physical descriptors, and equifinality in parameter estimation. These limitations have motivated the integration of data-driven modeling approaches, leading to increased interest in applying AI techniques to hydrological modeling (e.g., reviews in ).

In general, there are two main approaches for integration of machine learning (ML) and deep learning (DL) techniques in hydrological modeling. The first approach involves using a full black-box model that replaces traditional rainfall–runoff models. These purely data-driven models have been applied successfully to hydrological prediction, achieving state-of-the-art performance in various applications using long short-term memory (LSTM) network and their variants like LSTM-based Seq2Seq model . For example, reported that across 241 catchments in the US, their LSTM model achieved a mean Nash–Sutcliffe efficiency (NSE) of 0.63 in temporal validation, with over 50% of catchments reaching NSE values above 0.65.

A second approach is the hybrid method, which seamlessly integrates neural networks (NNs) into process-based numerical models, adhering to the principle of “learning under physical constraints.” The development of this approach has followed two independent trajectories. The first builds on recent efforts to make traditionally low-dimensional or lumped conceptual models differentiable and trainable end-to-end using modern automatic differentiation tools e.g.,, enabling the integration of NNs for tasks such as lumped-parameter regionalization or improved estimates of potential evapotranspiration for distributed hydrological modeling . A second trajectory stems from the long-standing use of adjoint state methods to compute gradients in high-dimensional, spatially distributed hydrological and hydraulic models, where the foundations of differentiability and numerical approximation were established well before dynamic-graph autodiff. Recent works have integrated NNs by coupling their Jacobians with adjoint-based gradients for gridded-parameter regionalization , including improvements via soil moisture assimilation , and for refining internal water fluxes . Additionally, merging process-based differential equations with ML can be highly advantageous. This has recently been demonstrated with physics-informed NNs in , where the process-based model serves as a weak constraint in the training cost function and is well-suited to assimilate observations e.g.,.

Despite the success of both pure and hybrid ML models, each approach faces several limitations and challenges. First, pure ML models are generally highly sensitive to large and high-quality training datasets, making them less reliable than process-based models in data-scarce regions . Unlike physics-based models, which rely on governing equations, ML models do not inherently impose physical constraints, resulting in reduced generalizability under extreme or unseen hydrological conditions . Hybrid modeling is a promising approach to address these limitations by objectively integrating ML/DL components into physically based models, rather than replacing them. However, hybrid approaches also face their own set of challenges. A primary limitation is maintaining physical consistency throughout the hybrid framework while leveraging the flexibility of data-driven approaches . Moreover, the integration of ML components with physical models may introduce complexities in model coupling and error propagation across different components . Current hybrid approaches are predominantly employed for lumped models, and hybrid modeling at high spatio-temporal resolution has received little attention, likely due to model complexity (with coupled processes) and the challenges of optimizing high-dimensional parameters (with gridded conceptual parameters) over large domains at high-resolution.

High-resolution hydrological modeling incorporates spatially distributed information at relatively fine temporal scales, such as gridded radar rainfall, essential for representing extreme flood events characterized by strong variability and short duration. However, such resolutions also introduce challenges for model calibration due to the high dimensionality of optimization parameters. Although this high-dimensional calibration problem can be tackled with a variational data assimilation (VDA) framework using numerical adjoint models of spatially distributed differentiable hydrological models , one is still facing overparameterization issues due to the sparsity of constraining calibration data such as in situ discharge timeseries, compared to large vectors of spatialized parameter. This problem can be addressed by introducing stronger constraints into the forward model, such as learnable mappings between physical descriptors and conceptual parameters . In addition, challenges remain due to both input data uncertainty and structural uncertainty in the hydrological model.

Another important point is that the reservoir states in conceptual hydrological models can be computed continuously and simultaneously by numerically solving a system of ordinary differential equations (ODEs) rather than sequentially (e.g., with state-space GR4 model). Continuous state-space hydrological models (i.e., those formulated within a dynamic system) typically employ implicit numerical ODE solvers to integrate the system's evolution over time. Indeed, under the time-step and forcing conditions tested by , fixed-step explicit schemes produced unacceptably poor hydrological simulations – even for parameter sets yielding good performance – whereas implicit schemes maintained numerical stability and physical consistency. An adjoint-based implicit scheme (designed to simplify the Jacobian computations required for implicit resolution) was proposed for a hybrid lumped model by , in which NNs were employed solely for parameter regionalization.

Recent studies indicate that embedding NNs into differential equation-based models, where the NN acts as a functional approximator or correction for selected source terms within a physically based differential equation offers a promising direction for advancing process representation in complex geophysical systems or biological systems . This generic framework, referred to in the scientific ML community as universal differential equations (UDEs, ), provides a flexible approach for integrating prior mechanistic knowledge with data-driven components in differential models. For instance, in hydrological modeling, implemented NNs that correct or replace precipitation-related source terms without dependence on the model states in the ODE system, and solved the system using an explicit numerical solver. However, to represent physical process memory, feedbacks, and nonlinear state interactions, such NN components should ideally depend on hydrological states, and the ODEs should be solved using implicit numerical schemes as discussed above. Thus, a rigorous numerical approach for solving UDE system – including NNs that depend on the hydrological model states – using implicit time integration schemes remains unexplored in hydrological modeling. Key challenges include the efficient computation of the Jacobian matrix for state-dependent NNs within the UDE system for its resolution, and the derivation of a numerical adjoint of the complete hydrological model including UDE and gridded kinematic wave (partial differential equation, PDE) routing to enable high-dimensional parameter optimization.

This study proposes a hybrid physics–AI approach that integrates state-dependent NNs within UDEs, into a differentiable, regionalizable, and spatially distributed hydrological model, where the UDE system is solved using an implicit numerical scheme. With the term “hybrid physics–AI,” we refer to an approach that seamlessly integrates NNs directly into the physical differential model and its numerical solver. This differs from approaches that treat NNs separately, for example, as pre- or post-processing components. The approach implements a mathematically rigorous method for computing the Jacobian matrices required by the implicit scheme when incorporating NN components. While we initially focus on simpler NN architectures to ensure numerical stability and physical interpretability, the hybrid numerical solver is designed to be compatible with PyTorch's automatic differentiation and could employ more complex architectures or deeper networks for process-parameterization.

The key novelty of this work lies in its capacity and generalizability to solve a spatialized UDE system with an implicit numerical scheme. This UDE system includes a NN that depends on the model states to refine internal water fluxes (source terms in the right-hand side of the ODEs), which leverages the dynamical system's internal state variables to retain information about past forcings and responses, thereby inducing temporal dependencies analogous to those captured by recurrent NNs (RNNs). This design allows the model to represent temporal memory effects and learn corrections for structural errors in the conceptual hydrological model. Additionally, the regionalization NN from is extended to explore alternative NN architectures, such as convolutional NNs (CNNs) in addition to multilayer perceptrons (MLPs), to improve the adaptability and scalability of parameter estimation. These developments aim to bridge the gap between ML flexibility and physical model interpretability, uncovering hydrological behaviors and scale-relevant theories inferred with AI techniques. It enables the addressing of several open research questions by providing a robust and powerful tool for enhancing flood modeling, mitigating structural uncertainty in modeling, optimizing data efficiency, and enabling more effective multi-scale information extraction through hybrid flux correction.

This paper also introduces smash v1.1, an upgraded version of the smash platform, following its initial release v1.0 in . The new version includes various hybrid physics–AI hydrological solvers and provides a more comprehensive user guide, along with detailed mathematical descriptions of the implemented models (see smash~1.1.0 Release Notes (https://smash.recover.inrae.fr/release/1.1.0-notes.html, last access: 5 November 2025) for details). As the core solver for the French flash flood forecasting system, smash is positioned to improve real-world flood simulation and hydrological forecasting, facilitating the integration of AI-enhanced physics-based modeling into operational hydrology.

Let us first define the domain, the main quantities of interest, and the general formulation of the dynamic hybrid model along with its dependencies. Consider a two-dimensional spatial domain Ω⊂R2, with x∈Ω as the spatial coordinate and t∈]0,T] as the physical time. Here, a regular grid is assumed, with an 8-direction (D8) surface flow path drainage network, DΩ. The hybrid rainfall–runoff model M (Eq. ) dynamically maps atmospheric forcings I(x,t)=[P,E](x,t) onto state variables of surface discharge Q(x,t), internal states h(x,t), and internal fluxes q(x,t), depending on learnable spatio-temporal correction fq(x,t) applied to internal fluxes q(x,t), on spatialized physical parameters θ(x) and initial states h0(x) that can be inferred through learnable regionalization mappings. 1Q,h,q(x,t)=MDΩ,I(x,t);fq(x,t),θ,h0(x).

The hybrid spatially distributed hydrological model M is constructed by chaining differential equations and NNs, consisting of: (i) a dynamic hydrological component Mrr, operating at the pixel scale, that simulates moisture states h, internal fluxes q, and surface discharge Q. This component is formulated as a UDE system, made learnable through a set of NN operators ϕ that provide spatialized hydrological parameters θrr and spatio-temporal flux-corrections fq; and (ii) a dynamic and gridded hydraulic routing module Mhy that transports surface discharge Q, whose parameters θhy can also be inferred by a regionalization NN. Together, these components, operating on the same spatial grid, define the forward model that is written, ∀x∈Ω,t∈]0,T], as in Eq. (). 2M=Mrr-hy(⋅;ϕ(⋅)):Hydrological operator Mrr (steps (0)–(2)):(0) Interception:PnEn=Fitc(P,E,ci)(1) UDE dynamics (GR4-like):dhdt=dhpdtdhtdt=(1-(hpcp)α1)Pn(1+fq,1[h])-hpcp(2-hpcp)En(1+fq,2[h])0.9(hpcp)α1Pn(1+fq,1[h])+kexc(htct)α3(1+fq,3[h])-ctα2-1(htct)α2(1+fq,4[h])(2) Flux closure law:Qlat=0.1(hpcp)α1Pn(1+fq,1)+kexc(htct)α3(1+fq,3)+ctα2-1(htct)α2(1+fq,4)(3) Hydraulic routing PDE Mhy:∂xQ+akwbkwQbkw-1∂tQ=λQlat

Now, let us explain the above governing hydrological equations. First, an interception reservoir Fitc with a capacity of ci, automatically computed using the flux matching technique , allows for the computation of the neutralized rainfall Pn and neutralized evapotranspiration En (the neutralization of original rainfall and evapotranspiration by Fitc).

Then, the NN-based estimator ϕ which hybridizes the above differential equation system and consists of a pair of NNs, which takes as input (i) neutralized atmospheric In=[Pn,En](x,t), along with the hydrological model states h(x,t), including production state hp and transfer state ht, to correct spatio-temporal internal fluxes q(x,t) (process-parameterization pipeline); and (ii) physical descriptors D(x) to estimate spatialized hydrological parameters θ(x) (regionalization pipeline), as shown in Eq. (). 3ϕ:fq(x,t)=ϕfluxIn(x,t),h(x,t);ρfluxθ(x)=ϕregioD(x);ρregio with ρ=(ρflux,ρregio) the vector of trainable parameters of the NN-based operator ϕ. The vector of conceptual parameters of the chained hydrological-hydraulic model that we aim to regionalize is defined as θ(x)=(θrr(x),θhy(x))T, here θrr(x)=(cp(x),ct(x),kexc(x))T and θhy(x)=(akw(x),bkw(x))T, where cp [mm] is the capacity of the production reservoir, ct [mm] is the capacity of the transfer reservoir, kexc [mmdt-1] is the non-conservative water exchange parameter, and akw,bkw [–] are the parameters of the kinematic wave routing. In this study, the NNs considered include either two MLPs, or an MLP for water flux correction combined with a CNN for parameter regionalization. Details on the NN architectures are provided in Appendix .

Next, the hydrological states h are computed by solving the UDE system of Eq. (), which can be generally expressed as: 4dhdt=Fgr(.,h,ϕflux(.,h;ρ)) where the left-hand side represents the dynamic evolution of the system and the right-hand side is the source term that defines the dynamic, here a GR4-like operator hybridized with the following two key components:

The NN-based operator ϕflux takes the hydrological model states as part of its inputs (a so-called state-dependent NN), thus affecting the model dynamics and state gradient information. It is expected to learn the model behavior by leveraging memory effects through state updates.

In this study, we consider Fgr in Eq. () to be a set of GR production and transfer operators (UDE system of Eq. ), where α1=2, α2=5, α3=3.5 are classical GR constants (cf. ); cp, ct, and kexc represent the conceptual parameters predicted by the NN ϕregio; fq,i=1..4 are the corrections applied to the internal fluxes, predicted by the NN ϕflux. The bracket notation [h] in the UDE system of Eq. () indicates that each flux correction functionally depends on h, implicitly encoding the system's memory of past forcings and responses. The physical constraints are enforced by the UDE system that underlies the hydrological state-space model and can be flexibly replaced by alternative physical laws within the proposed framework.

Note that mass conservation and non-conservative exchange fluxes have been further investigated and analyzed over a large sample using an algebraic resolution of the ODE system in . The closure relation in Eq. () follows a simple flux summation under the GR-like hypothesis at each pixel (for a detailed algebraic formulation, see and ). The numerical resolution of the hydrological UDE in Eq. (), ensuring both accurate resolution and numerical differentiability with respect to model parameters for optimization, will be explained in Section .

Finally, the routing module utilized here is based on a conceptual 1D kinematic wave model, which is numerically solved using a linearized implicit numerical scheme . Typically, the discharge routing problem is simplified to a 1D problem by adopting a “D8” drainage scheme DΩ(x), derived from processing a digital elevation model (DEM) of the terrain, with the assumption that a single pixel exhibits the largest drained area. The kinematic wave model is a PDE obtained by simplifying the 1D Saint-Venant equations, assuming the momentum is reduced to the flow friction slope, which equals the bottom slope. This is done by employing a conceptual parameterization for the momentum, A=akwQbkw, where A represents the flow's cross-sectional area, Q is the discharge, and akw and bkw are parameters to be estimated by the NN ϕregio. This expression is inserted into the mass equation ∂xQ+∂tA=λQlat, with Qlat being the lateral discharge (total runoff generated at a pixel from the GR operators described above), and λ representing the conversion factor. The result is a single-equation discharge propagation model as shown in the PDE of Eq. (). This kinematic wave model is numerically solved with a classical finite differences approach (cf. ). The routing solver is implemented in a numerically differentiable form after the hydrological module, allowing derivation of a numerical adjoint for the full hydrological–routing chain and thereby enabling end-to-end gradient-based parameter optimization.

The following section will detail the numerical method used to solve the UDEs (or the standard ODE system in the absence of a state-dependent NN) in Eq. ().

To solve Eq. (), we employ an implicit Euler scheme. For a small time step dt, by defining h˙=dhdt, we have: 5h(t+dt)=h(t)+dt⋅h˙(h(t+dt))

Now, define: 6gh(t+dt)=h(t+dt)-h(t)-dt⋅h˙h(t+dt)

Approximating the sought state h(t+dt) thus reduces to numerically solving the equation: 7g(y)=y-c-dt⋅h˙(y)=0 where y=h(t+dt) and c=h(t). Then, the solution of Eq. () is approximated using the Newton–Raphson method as follows: 8y0=c,yn+1=yn+Δy,where Δy is the solution of ∇g(yn)⋅Δy+g(yn)=0

The Jacobian matrix ∇g is given by: 9∇g=1-dt∂hp˙∂hp∂hp∂ht-dt∂hp˙∂ht∂ht∂hp-dt∂ht˙∂hp1-dt∂ht˙∂ht≈1-dt∂hp˙∂hp-dt∂hp˙∂ht-dt∂ht˙∂hp1-dt∂ht˙∂ht

This simplification holds because ∂hp∂ht=0 since production does not depend on transfer. Additionally, we assume that ht depends on hp only through its time derivative rather than instantaneously. Thus, ht evolves as an accumulated effect of hp, meaning its dependence on hp is indirect and primarily through its time derivative, justifying the approximation ∂ht∂hp≈0. The remaining terms, which are ∂hp˙∂hp, ∂hp˙∂ht, ∂ht˙∂hp and ∂ht˙∂ht, can be derived analytically with distinct formulations depending on whether the process-parameterization NN ϕflux is included, as follows.

First, we introduce a variable change: h̃=(hp̃;ht̃)T=(hpcp;htct)T. Since the conceptual parameters cp and ct remain constant over time, their derivatives vanish, leading to: dhp̃dt=1cpdhpdt and dht̃dt=1ctdhtdt. Then, the Jacobian matrix ∇g in Eq. () can be computed as follows in two different cases:

For classical continuous state-space structure (ODE): The process-parameterization NN is absent, we set fq≡0 in Eq. (). The resulting Jacobian components are:10∂hp˙∂hp̃=-α1Pnhp̃α1-1-2En(1-hp̃)∂hp˙∂ht̃=0∂ht˙∂hp̃=0.9α1Pnhp̃α1-1∂ht˙∂ht̃=α3kexcht̃α3-1-α2α2-1ctht̃α2-1

Finally, convergence is defined in terms of the relative Newton update measured in the Euclidean (ℓ2) norm: 12Δhh2<10-6 with a maximum of 10 Newton–Raphson iterations per time step.

The following section describes how the NNs – for both regionalization and flux correction – are trained with gradients propagated seamlessly through the entire modeling chain, including the hybrid hydrological module and the simplified hydraulic routing.

Considering the observed and simulated discharge time series Q∗=(Qg=1..NG∗)T and Q=(Qg=1..NG)T, where NG is the number of gauges across the study domain Ω, the discrepancy between the model and multi-catchment observations is evaluated using a cost function J as follows: 13J(Q∗,Q)=∑g=1NGwgj(Qg∗,Qg)

Here, ∑g=1NGwg=1 (with wg=1/NG in this study), and j(Qg∗,Qg)=1-NSE(Qg∗,Qg) for each gauge. Thus, J is a convex and differentiable function that relies on the output Q of the forward model M, and consequently depends on the conceptual parameters θ and the flux correction fq, and therefore on the parameters ρ of the NNs (cf. Eq. ). The VDA optimization problem is defined as shown in Eq. (), which involves optimizing the weights and biases of the NN-based operator ϕ. 14ρ^=arg⁡min⁡ρJQ∗,Mrr-hy(.;ϕ(.y;ρ))

Note that the entire forward modeling chain – including runoff production (whether solved algebraically, with a classical ODE solver, or a UDE solver), the kinematic wave routing module, and all NN components – is fully differentiable, which enables seamless gradient backpropagation. To address the inverse problem in Eq. (), we use the Adam optimizer, a gradient-based algorithm with an adaptive learning rate suitable for non-smooth objective functions. The optimizer requires the gradient ∇ρJ≡(∇ρfluxJ,∇ρregioJ)T to update the weights of ϕ. These gradients are obtained by solving the numerical adjoint model obtained by automatic differentiation with the Tapenade engine , which can also be combined by chain rule to the jacobian of an external regionalization NN as introduced by , enabling end-to-end training of the hybrid models. The NNs are trained as follows:

Pre-calibration of conceptual parameters: In the first step, we pre-train only the regionalization NN ϕregio to find a spatially distributed first guess for the conceptual parameters. The weights of the process-parameterization NN ϕflux, which is embedded in the ODE solver, are initialized using a normal distribution centered at zero with a small variance. This ensures that, while the non-zero initialization allows for proper flow of gradients in the network, the outputs of ϕflux being close to zero result in minimal corrections, thus preserving the original hydrological model structure. In other words, ϕflux has very limited impact during this phase. The pre-training uses a relatively high learning rate of 0.004 for faster gradient descent over 40 iterations. In the case of lumped parameters (i.e., ρ=θ‾ and ϕregio≡idϕregio), we simply use a heuristic optimization algorithm to provide a spatially uniform first guess.

It is worth noting that the embedded network ϕflux must be twice differentiable mathematically (once for solving the ODEs or UDEs, and once for the calibration process). While many activation functions in NNs are not differentiable at zero, such as ReLU (Rectified Linear Unit), stochastic optimization can often bypass this problem since the network generally produces non-zero (or non-close-to-zero) values. However, as we aim to preserve the original structure by producing outputs close to zero during pre-calibration, numerical errors can arise. To address this issue, we use the SiLU (Sigmoid Linear Unit) activation function in the hidden layers, as it is twice differentiable everywhere and provides smooth gradients.

The models are run at a spatial resolution of 1 km and an hourly time step. The methods are evaluated using a national database covering Metropolitan France with multi-source data. The study area is the Aude River basin, located in southern France, as in , but with a coarser spatial resolution of 1 km instead of 500 m. This area covers approximately 10 400 km2 (corresponding to a 104 × 100 grid), with an active domain of 4902 km2 (4902 active cells), and comprises 25 catchments, including 12 upstream gauges and 13 downstream gauges (Fig. ). The study period spans 9 years, from August 2014 to July 2023, divided into two sub-periods: P1 (2015–2019) and P2 (2019–2023). We use additional one-year warm-up periods (2014–2015 for P1 and 2018–2019 for P2) to calibrate or validate the models. Period P1 is used for calibration, while P2 is used for validation.

We perform calibration across multiple catchments using upstream gauges and evaluate regionalization performance by transferring parameters to downstream gauges, which represents a more challenging regionalization case compared to transferring parameters from downstream to upstream gauges. A set of seven descriptors with a spatial resolution of 0.01° in the WGS 84 projection, similar to , is used as inputs for the regionalization mapping ϕregio (Fig. ).

We evaluate the models based on two criteria: model structure flexibility and regionalization ability. Model structure flexibility is assessed by the model's capacity to produce interpretable water flux dynamics, both with and without the process-parameterization network ϕflux. Regionalization ability is evaluated based on the capability of the NNs ϕregio, which can be either an MLP or a CNN, to regionalize the conceptual parameters using physical descriptors. Table  summarizes the evaluated models and indicates the version of smash in which each was first introduced.

Figure  illustrates the forward hydrological model through a schematic representation of the different evaluated models. The computational cost of these models is reported in Appendix .

A major advantage of numerically solving ODEs, as opposed to algebraic resolution methods, lies in their generalizability. While an analytical solution derived through an algebraic approach is exact, it can only be obtained under specific assumptions about the ODE system (e.g., certain fixed values of the coefficients (αi)_i=1..3 in Eq. ). In contrast, numerical methods can solve ODEs without requiring such assumptions, albeit with approximate solutions. It is thus necessary to validate the ODE solver (ODE solutions obtained by numerical scheme) against the algebraic approach before performing any numerical experiments based on this solver. This is particularly important as this work presents the first implementation of both classical and hybrid ODE solvers into a fully distributed hydrological modeling and VDA framework.

To carry out this validation, we compare the GR algebraic model with lumped parameters (GR.U), which solves an analytical solution of the time-integrated ODEs, and the continuous state-space model also with lumped parameters (ODE.U), which solves the ODEs using an implicit numerical scheme. For both models, we set identical conceptual parameters and initial states, which are non-calibrated. Figure  shows similar simulated hydrological responses for GR.U and ODE.U obtained at the most downstream gauge. Although slight differences are found in the production state (often higher state for ODE.U) at certain periods, the transfer state for both models remains nearly identical, while streamflow simulation shows minor differences at several peak flows.

In addition to similar temporal hydrological responses, Fig.  demonstrates nearly identical spatial patterns and values of the final model states obtained for both methods. The bias map between ODE.U and GR.U for the production state shows small deviations centered around zero, while the bias for the transfer state is nearly negligible, aligning with the earlier temporal observations. These minor discrepancies between the two approaches can be attributed primarily to the fundamental difference in their solution methodology – namely, the simultaneous numerical resolution of the ODEs for both state variables (hp and ht) versus the exact analytical solution obtained through sequential algebraic resolution – and approximation errors of the Newton–Raphson method.

While both methods initially produce similar hydrological responses, it will be particularly interesting to observe in the next section how the model dynamics evolve during calibration, and whether differences in the numerical formulation influence model behavior and performance.

Figure  provides a global view of model performance across calibration and various validation scenarios. Results demonstrate that model scores in calibration increase with the complexity of regionalization mappings (from uniform to MLP and CNN). For each regionalization approach, we observe improved scores progressing from classical GR structure to ODE and UDE structures, exemplified by a median NSE over 0.85 for UDE.CNN compared to 0.82 for ODE.CNN and 0.8 for GR.CNN. Moreover, models with ODE and UDE structures show improved interquartile ranges and whiskers when using CNN or MLP regionalization compared to GR, while UDE.U exhibits larger variance despite maintaining higher median scores than GR.U.

For temporal validation, ODE.MLP and UDE.MLP yield similar high median scores of 0.71 compared to 0.59 for GR.MLP. Additionally, UDE.MLP demonstrates a superior interquartile range, with a 0.25-quantile of 0.65 compared to 0.57 for ODE.MLP and 0.42 for GR.MLP. Models with the same regionalization approaches (uniform or CNN) show similar performance across different structures (GR, ODE, UDE), with only slight differences in median and distribution.

Spatial validation shows comparable performance across structures for each mapping, except for ODE.CNN which performs relatively worse than GR.CNN and UDE.CNN. While all models struggle with spatial validation, spatio-temporal validation yields generally good performance overall. Although ODE.CNN maintains lower performance compared to GR.CNN and UDE.CNN, the MLP-based models exhibit a more stable performance across validation scenarios. For instance, ODE.MLP achieves the best median scores among the three MLP-based models for spatio-temporal validation, while UDE.MLP yields the best interquartile range. Interestingly, GR.CNN exhibits promising performance in spatio-temporal validation with a median score of 0.71.

Overall, GR.CNN and UDE.MLP demonstrate consistently strong and stable performance across all calibration and validation scenarios. While GR.CNN generally outperforms its MLP counterpart in validation scenarios, UDE.MLP shows stable performance with good scores and favorable interquartile ranges across all validation scenarios.

To illustrate the models' flood simulation capabilities, Fig.  presents two representative flood events during the validation period, as observed at an upstream and a downstream gauge. At both locations, all models using uniform mapping significantly underestimate the flood magnitudes, resulting in the poorest simulation performance. This result highlights the importance of regionalization approaches in accurately simulating floods, particularly in ungauged basins, where traditional methods that use lumped parameters often fail to capture real hydrological dynamics. For the upstream gauge example, regionalization approaches using MLP and CNN produce relatively similar discharge simulations for each model structure (GR, ODE, UDE). The ODE and UDE structures with MLP show slightly better performance, with RMSE values of 4.37 and 4.13, compared to 6.2 for GR.MLP. Both CNN and MLP mappings applied to the GR structure accurately simulate the peak flow, whereas the regionalization methods for the ODE and UDE structures tend to slightly overestimate peak flow, particularly in the case of ODE.CNN. Despite this, the timing of flood events remains accurate across all regionalization methods, with the rising limbs of the simulated hydrographs closely aligning with observations. Note that in this upstream catchment characterized by a quick hydrological response, the start of the flood event occurs nearly at the same time as the moment of heavy rainfall. In contrast, the downstream gauge event shows an approximately 8 h delay between heavy rainfall and flood response. Most models struggle to predict the lag time correctly. This emphasizes the need to improve model realism by accounting for rainfall intensity and studying its impact in triggering non-linear flash flood responses, improving river network hydraulics, which presents a promising avenue for future research. Returning to flood simulation at the downstream, while the majority of models yield poor performance in predicting the event's beginning, generating flood responses earlier than observed, GR.CNN demonstrates impressive timing accuracy with simulations very close to observations. However, GR structure models, including GR.CNN, still underestimate flood magnitude, whereas ODE.CNN, UDE.MLP, and UDE.CNN perform more accurately.

Figure  shows the maps of spatially distributed conceptual parameters calibrated using CNN and MLP regionalization approaches. Overall, CNN-based models produce smoother parameter maps compared to MLP-based models, as evidenced in the maps of cp and ct for GR.CNN (compared to GR.MLP), kexc for all CNN-based structures, akw for all CNN-based structures, and bkw for GR.CNN. This smoothing effect results from the convolution operations applied to physical descriptor maps. Furthermore, we observe spatial patterns from physical descriptors (Fig. ) reflected in the parameter maps, such as the pattern at the top of the forest cover map (d4) or the pattern at the bottom left of the slope (d1). For each state-space structure, different regionalization mappings produce distinct parameter distributions (e.g., different patterns of GR.MLP compared to GR.CNN, ODE.MLP compared to ODE.CNN, and UDE.MLP compared to UDE.CNN). Additionally, notable differences in parameter patterns emerge across model structures, with major differences for GR-based models, while ODE and UDE structures yield relatively similar parameter maps (ODE.MLP compared to UDE.MLP, and ODE.CNN compared to UDE.CNN). This pattern divergence is expected since classical GR models have different state dynamics compared to continuous state-space models, even when employing the same regionalization mapping.

Figure  examines model dynamics through the spatial average of normalized states (production state hp and transfer state ht). The two continuous state-space models (ODE.MLP and UDE.MLP) demonstrate similar behavior in production state, showing higher values and lower variability compared to GR.MLP. However, for the transfer state, the UDE.MLP model produces higher values and greater variability compared to the ODE.MLP and GR.MLP models. This likely results from the hybridization effect of the process-parameterization NN ϕflux, which incorporates neutralized rainfall to refine the physical equations in the ODE system. This creates a rainfall sensitivity in the transfer state, which is not accounted for in the classical GR and ODE structures.

Nevertheless, Fig.  reveals that all three model structures exhibit relatively similar patterns in time-averaged transfer state maps, though with higher mean values (0.18 compared to 0.16 for ODE.MLP and 0.17 for GR.MLP) in the case of UDE.MLP. The maps of time-averaged production state for UDE.MLP and ODE.MLP display different patterns from those of GR.MLP. However, no evident differences emerge between ODE.MLP and UDE.MLP for both hp and ht. This is unsurprising as spatial hybridization effects are not expected since the process-parameterization NN employs a simple MLP (MLP of ϕflux and not MLP of the regionalization mapping ϕregio) that does not account for or accounts less for spatial information. Future work could employ other types of ϕflux to explore this aspect.

Finally, Fig.  illustrates the hybridization effect on runoff flow (lateral discharge for routing operator) by examining high-pass filtered lateral discharge from the most downstream gauge during the validation period. A 3 d cutoff frequency high-pass filter removes seasonal and long-term patterns to focus on flood event behavior. During major events, GR.MLP consistently produces lower lateral discharge compared to ODE.MLP. In certain situations, this results in an overestimation of flood magnitude with the ODE structure and an underestimation with the GR structure. The UDE structure effectively addresses this issue by refining internal water fluxes in the ODE system, resulting in moderate lateral discharge values that fall between those of ODE.MLP and GR.MLP. This effect is consistent with previously observed flood simulation results, demonstrating the improved hydrological response representation achieved through UDE integration.