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REVIEW 3 major objections 5 minor 1 cited by

MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read MultiPDENet, a PDE-embedded network with multi-scale time stepping, predicts long-term flow dynamics accurately on coarse, heavily downsampled grids from as few as three to five training trajectories, with a five- to seven-fold speedup…

desk verdict A well-ablated hybrid solver with a genuinely novel multi-scale coupling, but the 'long-term' accuracy claim is only validated up to the 8.4 s test horizon. read the letter →

arxiv 2501.15987 v1 pith:XF3I56QH submitted 2025-01-27 math.NA cs.AIcs.NA

Qi Wang , Yuan Mi , Haoyun Wang , Yi Zhang , Ruizhi Chengze , Hongsheng Liu , Ji-Rong Wen , Hao Sun This is my paper · ORCID
classification math.NAcs.AIcs.NA MSC 65M0668T0776D05
keywords PDE-embeddedlearningmulti-time-steppingcoarse-gridsimulationphysics-encodedneuralnetworkKolmogorovflowclosuremodelingRunge-Kuttaintegratorsymmetricconvolutionfilter
topics Navier-Stokes Regularity
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that a neural network can simulate flows accurately and cheaply on grids far coarser than exact numerical solvers need, provided the governing PDE is embedded in the architecture itself rather than merely added as a loss term. MultiPDENet does this with a two-scale time-stepping scheme: fine steps are taken by a learnable fourth-order Runge-Kutta integrator whose residual block computes the PDE's terms through trainable symmetric derivative filters, and coarse steps are corrected by a neural network that repairs the error accumulated over many fine steps. Trained on just three to five spatiotemporally downsampled trajectories, the model reports long-horizon predictions across the KdV, Burgers, Gray-Scott, and Kolmogorov-flow Navier-Stokes systems, reaching an RMSE of 0.1379 and a high-correlation time of 8.36 seconds on a 64x64 grid, compared with 0.2674 and 6.5 seconds for a 512x512 direct numerical simulation. If the claims hold, fine-grid DNS could be replaced for many purposes by a small, data-efficient surrogate that is also interpretable, because the embedded PDE block is a white-box component rather than a black-box fit.

What carries the argument

The load-bearing object is the multi-time-stepping scheme itself, Eq. (2), together with the Physics Block that implements it. The Physics Block is a white-box neural solver: a PDE block evaluates the residual of the governing equation using a symmetrically constrained, learnable filter bank (two $5\times5$ kernels, six trainable parameters each) that approximates the coarse-grid 'equivalent' of the spatial derivatives, and feeds it into a fourth-order Runge-Kutta integrator so that each micro step obeys the equation's structure and, through periodic padding, the boundary conditions. A Poisson block computes pressure spectrally for incompressible flow. The MaNN block then acts as a coarse-scale corrector, absorbing the error the RK4 rollout accumulates over many micro steps; the ablation study shows that removing it destroys long-term stability (Model G fails with NaN), while removing the Physics Block entirely (Model H) raises RMSE by an order of magnitude. The scheme's effect is to separate fast, equation-governed physics from the slow accumulation of closure error, letting each be handled by the component best suited to it.

What would settle it

Take the trained 64x64 model to a regime beyond the tested range where the closure assumption is strained, for instance Kolmogorov flow at Re = 10,000 on the same grid, a non-periodic geometry such as a lid-driven cavity, or a stochastic PDE with unresolved noise. If RMSE then grows without bound, or high-correlation time falls to the level of the DNS-64 baseline (about 2.7 seconds), the central closure premise is refuted. A sharper version: estimate the Lyapunov or predictability time of the Kolmogorov flow at the reported parameters; if MultiPDENet's 8.36-second HCT exceeds that intrinsic predictability limit, the favorable comparison with DNS would need re-examination against a resolution-dependent baseline.

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Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the equation form of a PDE carries enough inductive bias that a network built around it can recover the subgrid dynamics lost when data is downsampled by up to 1024x in space and 128x in time, using only 3-5 training trajectories. The prediction step $u^{k+1} = u^k + \sum_{m=1}^{M}\delta\overline{u}^k_m + \mathrm{MaNN}(u^k,\Delta t,dx)$ combines three learned corrections: a Physics Block that advances the coarse state through $M=4$ micro steps of RK4 applied to the embedded PDE residual, with a MiNN neural block refining each micro step; a symmetric $5\times5$ convolution filter bank with only six free parameters per derivative order that learns the 'equivalent form' of $\nabla$ and $\nabla^2$ on the coarse grid, constrained by order-of-sum rules so that derivative accuracy up to fourth order is reachable; and a MaNN block, a U-Net, that corrects the macro-step output and is shown by ablation to be the component that prevents long-term drift. For incompressible Navier-Stokes, pressure is obtained by solving the Poisson equation $\Delta p = \psi(u)$ spectrally, so no labeled pressure data is needed. The reported consequence is that the coarse-grid model is both more accurate and longer-lived than DNS at medium resolution, and generalizes to unseen initial conditions, Reynolds numbers from 500 to 4000, changed forcings, and doubled domain sizes without retraining.

Load-bearing premise

The load-bearing premise is that the coarse-grid state, together with the embedded PDE form and the learned corrections, contains enough information to reconstruct the subgrid dynamics removed by up to 1024x spatial and 128x temporal downsampling; if that small-scale information is genuinely absent from the coarse state, the RK4 rollout will drift and the long-term accuracy claim collapses.

Editorial extensions

If this is right

  • Simulation cost drops five- to seven-fold: producing 8.4 seconds of Kolmogorov flow with correlation above 0.8 takes 26 seconds on a single A100 GPU with MultiPDENet, versus 130-135 seconds for DNS at comparable accuracy.
  • Training data requirements collapse to three to five trajectories, and the model keeps working when 20% of snapshots are removed or 0.1% Gaussian noise is added, with RMSE rising only to 0.19-0.21 while correlation time stays above 8 seconds.
  • The same trained weights transfer to unseen initial conditions, Reynolds numbers from 500 to 4000, four different forcing terms, and a doubled spatial domain, with error distributions below 0.1 and correlations matching or exceeding DNS at 1024-4096 resolution.
  • Because the PDE block is white-box, the learned symmetric filters supply an interpretable coarse-grid form of the differential operator, which matters when the PDE parameters $\lambda$ are unknown and set trainable.
  • The ablations attribute the gain: the MaNN macro-corrector is the component that keeps long-term rollouts stable (removing it yields NaN), the symmetric filter constraint beats unconstrained or fixed finite-difference kernels, RK4 beats forward Euler, and the Poisson block measurably improves incompressible-flow accuracy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The architecture is effectively a learned closure model: the trainable filters plus the macro correction estimate subgrid fluxes that a coarse-grid solver cannot resolve. A natural step the paper does not take is swapping the RK4 core for a finite-volume or spectral base discretization, which would test whether the closure idea rather than the specific stencil carries the accuracy.
  • The reported transfer across Reynolds numbers hints that the learned derivative operators encode a Reynolds-number-independent structure; a testable prediction is that a filter bank trained on forced Kolmogorov flow should partially transfer to a different flow with the same differential structure, such as Rayleigh-Benard convection.
  • The long-horizon claim carries an implicit assertion about predictability: a coarse state must determine evolution beyond the decorrelation time of small scales. Comparing MultiPDENet's high-correlation time with the Lyapunov time of the Kolmogorov flow at each Reynolds number would mark the accuracy ceiling for any grid-coarsening surrogate.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. MultiPDENet proposes a PDE-embedded neural architecture for accelerated simulation of spatiotemporal dynamics on coarse grids. The model combines a learnable symmetric finite-difference filter bank, a physics block with RK4 integration, a Poisson solver for incompressible flows, and a multi-time-stepping scheme in which a MiNN block corrects micro-scale increments and a MaNN block corrects macro-scale updates. The paper evaluates MultiPDENet on KdV, Burgers, Gray-Scott, and Kolmogorov Navier-Stokes flows using only 3–5 training trajectories, and reports that it outperforms FNO, UNet, DeepONet, PeRCNN, LI, and TSM baselines in RMSE/MAE/MNAD/HCT, and achieves 5–7x speedups over DNS at matched accuracy. Additional generalization experiments cover unseen initial conditions, Reynolds numbers (500–4000), forcing terms, and a larger (4π)^2 domain. Ablation studies identify the contributions of each module.

Significance. If the reported results hold, MultiPDENet is a meaningful step toward hybrid PDE-embedded models that are data-efficient and interpretable. The architecture is described in sufficient detail to be reproduced, the ablation study is systematic and supports the role of each component, and the generalization tests across Reynolds numbers, forcings, and domain sizes are a genuine strength. The small-data regime (3–5 trajectories) is an attractive feature. However, the evaluation has three important weaknesses that bear on the central claims: the long-term accuracy claim is only demonstrated up to a truncated 300-step horizon, the Re=4000 experiment is ambiguous about retraining, and the baseline comparisons lack statistical error bars, with several baselines reporting NaN. These issues need to be addressed before the claims of long-term prediction and state-of-the-art performance are fully supported.

major comments (3)
  1. [Section 4.2 / Table S7] The headline 'long-term' claim for the Navier-Stokes Kolmogorov flow is not supported beyond the evaluation horizon. For Re=1000, MultiPDENet reports HCT = 8.3566 s, which essentially equals the full test rollout of 300 macro steps (~8.4 s); the HCT metric as defined in Eq. (S15) accumulates time steps only until the end of the rollout, so it is saturated and cannot detect instability or divergence after that window. Since the training trajectories contain 1200 coarse steps (~33.6 s) and ground-truth DNS is available for longer times, the authors should either report rollouts extending well beyond 8.4 s (e.g., 600–1200 macro steps) or explicitly qualify the 'long-term' claim as holding only up to 300 macro steps. The Re=4000 result (HCT=6.4 s, Table S7) further shows that stability is not uniform, reinforcing the need for extended-horizon evidence.
  2. [Section 4.3 / Appendix D.3] It is unclear whether the Re=4000 experiment uses the model trained on Re=1000 or a model retrained on Re=4000 data. The main text states that the model was initially trained at Re=1000 and then tested on Re=500, 800, 1600, 2000, but the Re=4000 paragraph says 'with the experimental setup in Section 4.1' and 'After training, the model was tested...', which suggests a separate training run. If the Re=4000 model was retrained, the experiment is not a zero-shot generalization test and the 'without retraining' claim in the caption of Figure 5 does not apply as stated; if it was not retrained, this should be stated explicitly. Please clarify and, if retrained, present the Re=4000 result separately from the generalization tests.
  3. [Table 2 / Section 4.1] Several baselines diverged (NaN) under the reported default settings: DeepONet on KdV, UNet on GS, and LI/TSM on NSE. Without error bars or a per-seed breakdown, it is not possible to tell whether these failures reflect intrinsic limitations of the baselines or poor hyperparameter choices; the 'state-of-the-art' comparison in Table 2 is therefore weakened. The authors should report results over multiple random seeds (mean ± std) for all models, or at least for MultiPDENet and the best baseline in each case, and should document the hyperparameter tuning protocol used for each baseline.
minor comments (5)
  1. [Appendix E, Eq. (S15)] 'High Correction Time' should be 'High Correlation Time' to match the HCT definition.
  2. [Section 3.2.1, Eq. (2)] 'anddx' should read 'and dx' with a space; please fix the typo.
  3. [Section 4.1, loss function] The loss J(λ) states 'N denotes the number of batches, B the batch size', but the summation is over i=1..B and j=1..N; swap the definitions for consistency.
  4. [Figure 3] Labels such as 'UNetRef.' and 'Ref.' are hard to parse; please use a single clear legend for the reference solution and each baseline.
  5. [Table 1] The entries under 'Macro-step Rollout' and 'Micro-step Rollout' are ambiguous; clarify whether the numbers denote time steps or physical durations.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: the generalization tests are genuine, and the only self-citations are minor related-work pointers.

full rationale

The central derivation is self-contained against external benchmarks. MultiPDENet is trained with an MSE data loss against coarse ground-truth snapshots (Section 4.1: J(lambda) = 1/(BN) * sum MSE(H_hat_ij, H_ij)) and is then evaluated on unseen initial conditions, Reynolds numbers (Re = 500, 800, 1600, 2000, 4000), forcing terms, and a larger domain (Section 4.3 and Appendix D.3), so the reported RMSE and HCT values are not fitted quantities renamed as predictions. The Physics Block is an inductive bias, built from RK4 integration, symmetric trainable derivative filters, and a spectral Poisson solver, rather than a target defined in terms of the output; the ablation study in Table 3 shows that removing each component degrades performance, confirming that the components carry independent content. The HCT saturation at the 8.4 s evaluation horizon is a scope or correctness concern about the strength of the 'long-term' claim, not circularity: HCT is computed from the predicted rollout and is not an input to the model. Several works with overlapping authorship are cited (P2C2Net, PeRCNN, PhySR, and Liu et al. 2024), but they appear only as related-work or baseline pointers and do not justify any load-bearing premise, uniqueness claim, or ansatz. No equation in the paper reduces to its own input by construction.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the assumption that coarse-grid states, the known PDE structure, and learned corrections can jointly close the subgrid model. The free parameters are the learned stencil coefficients, the Re embedding, the neural weights, and the hand-chosen number of micro-steps. No new physical entities are introduced.

free parameters (5)
  • Symmetric filter bank coefficients (a1-a6 for first derivative, a1-a6 for second derivative) = Dataset-specific, learned during training
    The convolutional filters approximate coarse-grid spatial derivatives; their 12 coefficients are optimized to reduce the PDE residual and data loss. Section 3.2.3.
  • Reynolds number embedding vectors a and b = Trainable, learned
    Re is encoded as Re_emb = (1/Re)(a ⊗ b) and applied in PDE and MiNN blocks. Section 4.1.
  • PDE parameters λ = Could be set trainable; in experiments likely fixed to known values
    Equation (1) defines λ as PDE parameters; the paper states λ can be trainable if unknown.
  • Neural network weights (MiNN FNO, MaNN UNet, Correction FNO) = Millions of parameters, learned
    The correction blocks are trainable and are central to the macro-step update; Tables S5 and S6 give sizes.
  • Number of micro steps M per macro step = M = 4
    Chosen by hand (Section 3.2.1, Table 1); affects the balance between RK4 accuracy and NN correction.
assumptions (7)
  • domain assumption The governing PDE form is known and can be embedded as the PDE Block (Equation 4 approximates F in Equation 1).
    The method assumes the user can write the PDE structure (e.g., NSE with advection, diffusion, pressure, forcing) so that the network only learns coefficients, not the operator form.
  • domain assumption Coarse-grid states contain enough information for the correction blocks to recover subgrid dynamics (closure assumption).
    The central claim of accurate long-term prediction from downsampled data requires that lost high-wavenumber information is inferable from the coarse state and the embedded PDE structure.
  • domain assumption Periodic boundary conditions on regular grids hold for all test cases.
    The architecture hard-encodes periodic padding (Figure S2) and uses a spectral Poisson solver, which assumes periodicity.
  • domain assumption The spectral Poisson solver on the coarse grid gives an adequate pressure field for the incompressible NSE.
    Pressure is computed from the coarse velocity via FFT; if the coarse field does not resolve the vorticity source, the pressure gradient used in updates could be wrong.
  • standard math The symmetric filters can represent equivalent coarse-grid derivatives up to 4th order (from Long et al. 2018).
    Section 3.2.3 relies on the Order of Sum Rules to claim 4th-order derivative approximation; this is background theory from the cited work.
  • domain assumption RK4 with micro-step δt satisfying the CFL condition controls temporal error accumulation.
    Section 3.2.2 and Appendix B.3 assume the micro-step is small enough for RK4 stability.
  • domain assumption The dataset generation (spectral/FD/FV high-resolution simulation) provides accurate ground truth, and downsampling does not alias the dynamics past learnability.
    All training labels come from downsampled DNS; any systematic error in the reference data transfers to the model.

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Cite this review

Pith. "Pith review of MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation." pith.science (2026). https://pith.science/paper/XF3I56QH

@misc{pith2026250115987,
  author       = {Pith},
  title        = {Pith review of: MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XF3I56QH}},
  note         = {Machine review of arXiv:2501.15987}
}
read the original abstract

Solving partial differential equations (PDEs) by numerical methods meet computational cost challenge for getting the accurate solution since fine grids and small time steps are required. Machine learning can accelerate this process, but struggle with weak generalizability, interpretability, and data dependency, as well as suffer in long-term prediction. To this end, we propose a PDE-embedded network with multiscale time stepping (MultiPDENet), which fuses the scheme of numerical methods and machine learning, for accelerated simulation of flows. In particular, we design a convolutional filter based on the structure of finite difference stencils with a small number of parameters to optimize, which estimates the equivalent form of spatial derivative on a coarse grid to minimize the equation's residual. A Physics Block with a 4th-order Runge-Kutta integrator at the fine time scale is established that embeds the structure of PDEs to guide the prediction. To alleviate the curse of temporal error accumulation in long-term prediction, we introduce a multiscale time integration approach, where a neural network is used to correct the prediction error at a coarse time scale. Experiments across various PDE systems, including the Navier-Stokes equations, demonstrate that MultiPDENet can accurately predict long-term spatiotemporal dynamics, even given small and incomplete training data, e.g., spatiotemporally down-sampled datasets. MultiPDENet achieves the state-of-the-art performance compared with other neural baseline models, also with clear speedup compared to classical numerical methods.

Figures

Figures reproduced from arXiv: 2501.15987 by the authors.

Figure 1
Figure 1. Schematic of MultiPDENet for learning turbulent flows. (a), Model architecture. (b), Physics Block. (c), Learnable PDE block. in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Symmetric filter compute the pressure, we employ a spectral method (Pois￾son solver) based on ψ(u¯ k m) to calculate p¯ k m. As shown in Figure S1(b), this approach dynamically estimates the pressure field from the velocity inputs, removing the need for labeled pressure data. 3.2.3. ADAPTIVE FILTER WITH CONSTRAINT Traditional FD methods often yield inaccurate derivatives on coarse grids. To address this, we propose … view at source ↗
Figure 3
Figure 3. An overview of the comparison between our MultiPDENet and baselines, including predicted solutions (left), correlation curve (middle), and error distributions (right). (a)-(c) show the qualitative results on KdV. (d)-(f) show the qualitative results on Burgers. (g)-(i) show the qualitative results on GS. These PDE systems are trained with grid sizes of 64, 25×25, 32×32, respectively. and coarse training data [PITH_… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Comparison of MultiPDENet and baseline models on Kolmogorov flow with Re = 1000. (a) shows the evolution of predicted vorticity fields for reference, MultiPDENet and baselines, starting from the same initial velocities. (b) shows the correlation curve across 500 time s…
Figure 5
Figure 5. Figure 5: MultiPDENet can generalize to simulate different Reynolds numbers and external forcings without retraining. Vorticity snapshots predicted by MultiPDENet and ground truth at timestep = 300 (left), correlation curve over 600 timesteps (middle), error distribution (right)…

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

13 extracted references · 12 canonical work pages · cited by 1 Pith paper

  1. [1]

    The model hyperparameters are listed in Tables S5 and S6

    Additionally, we use the StepLR scheduler to adjust the learning rate by a factor of 0.96 every 200 steps. The model hyperparameters are listed in Tables S5 and S6. FNO. The architecture of the FNO network closely follows that presented in the original study (Li et al., 2021), with the main adjustment being the adaptation of its training methodology to an...

    work page 2021

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    The data is initially generated on a 1002 grid and subsequently downsampled to a 252 grid for use in numerical experiments

    and periodic boundary conditions over the spatial domain x ∈ [0, 1]. The data is initially generated on a 1002 grid and subsequently downsampled to a 252 grid for use in numerical experiments. The simulation timestep is set to dt = 1 × 10−3 seconds, with a total duration of T = 1.4 seconds. During the training stage, we employ five trajectories with ∆t = ...

    work page 1914

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    G.2. Computational Cost (Inference) Taking NSE as an example, we compared the inference time, RMSE, and HCT of MultiPDENet with the Direct Numerical Simulation (DNS) method across three cases. The comparison principle is based on the time required to simulate the same trajectory length (T = 8.4 s) under identical experimental conditions (a single A100 GPU...

    work page 2021

  4. [10]

    cov” refers to the covariance function, while “ σ

    Data volume Figure S5. Comparison of PeRCNN and MultiPDENet across various training set sizes on the Burgers equation. The x-axis represents data volume, defined as the product of trajectory timesteps and the number of trajectories (corresponding to 5, 8, 16, 32, and 64 trajectories, respectively). Dotted lines denotes the linear interpolation. E. Evaluat...

    work page 2023

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    The learning rate is established at 5 × 10−4, with a decay factor of 0.9 applied every 5000 steps

    along with the Adam optimizer. The learning rate is established at 5 × 10−4, with a decay factor of 0.9 applied every 5000 steps. The model is trained using a batch size of 16 over a total of 20000 epochs. PeRCNN. We maintain the standard architecture of PeRCNN (Rao et al., 2023). The optimization process is executed with the Adam optimizer and employs a ...

    work page 2023

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    We adopt the default network architecture and parameter settings for LI (Kochkov et al., 2021)

    LI. We adopt the default network architecture and parameter settings for LI (Kochkov et al., 2021). The optimizer used is Adam with β1 = 0.9 and β2 = 0.99. The batch size is configured to 8, along with a global gradient norm clipping threshold of 0.01. The learning rate is set to 1 × 10−3, and weight decay is configured to 1 × 10−6. 21 PDE-embedded Learni...

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    to serve as the ground truth. Different ICs are generated by introducing random noise into each component of the velocity field and subsequently filtering it to obtain a divergence-free field with the desired properties. For training, we utilize only five groups of labeled data with 4800 snapshots, while testing involves ten sets of trajectories. The mode...

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    B.2. Physics Block To accurately predict at the micro-scale step, we developed a neural solver called the Physics Block, ensuring stability, accuracy, and efficiency through adherence to the Courant-Friedrichs-Lewy (CFL) conditions (LeVeque, 2007). The Physics Block comprises three key components: the Poisson Block (Figure S1(a)-b)), the PDE Block (Figure...

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Show all 13 references
  1. [2005]

    Related work Numerical Methods

    11 PDE-embedded Learning with Multi-time-stepping APPENDIX A. Related work Numerical Methods. Numerical methods have been extensively applied to solve PDEs. Approaches such as FD (Thomas, 2013), FE (Zienkiewicz et al., 2005), and FV methods (Moukalled et al.,

    2013

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    With abundant labeled data, it is possible to train end-to-end models to predict solutions

    and computer vision (He et al., 2016), these techniques have also been applied to solving PDEs. With abundant labeled data, it is possible to train end-to-end models to predict solutions. Representative works include ResNet (Lu et al., 2018), CNN-based models (Bhatnagar et al....

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    Hybrid Learning Methods

    have explored the use of CNN as alternative spatial derivative operators for approximating derivatives and capturing the dynamics of interest. Hybrid Learning Methods. Hybrid learning methods combine the strengths of numerical approaches and NNs to improve prediction accuracy....

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    Brandstetter, J., Welling, M., and Worrall, D. E. Lie point symmetry data augmentation for neural pde solvers.arXiv preprint arXiv:2202.07643, 2022a. Brandstetter, J., Worrall, D. E., and Welling, M. Message passing neural pde solvers. In International Conference on Learning R...

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    and Graph-based models (Brandstetter et al., 2022b). Many notable neural operators (Lu et al., 2021; Li et al., 2021; Rahman et al., 2023; Bonev et al., 2023), which learn a mapping between functional spaces, enable the approximation of complex relationships in PDEs. While the...

    2021

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