Encoding physics to learn reaction-diffusion processes

Chengping Rao; Pu Ren; Qi Wang; Oral Buyukozturk; Hao Sun; Yang Liu

Source-linked AI summary

Encoding physics to learn reaction-diffusion processes

Chengping Rao, Pu Ren, Qi Wang, Oral Buyukozturk, Hao Sun, Yang Liu

arXiv:2106.04781v2Jun 9, 2021cs.LGmath.NAphysics.comp-ph

TL;DR

Many spatiotemporal systems lack complete governing PDEs, while existing physics-informed models impose physics softly and can remain opaque or weak at extrapolation and generalization. The paper introduces PeRCNN, which hard-encodes known physics into a recurrent convolutional architecture and combines it with sparse regression for PDE discovery. Across numerical experiments, the framework is reported to support accurate, robust, interpretable, and generalizable forward, inverse, modeling, and discovery tasks, with limitations for irregular meshes and non-polynomial governing equations.

Problem
Many underexplored spatiotemporal systems have uncertain or incomplete governing PDEs, while existing physics-informed models rely on soft constraints and black-box networks.
Method
PeRCNN hard-encodes known PDE structure, initial and boundary conditions, and numerical time integration into a recurrent convolutional network, with sparse regression for equation discovery.
Results
PeRCNN supports forward and inverse analysis, data-driven modeling, and PDE discovery with reported accuracy, robustness, interpretability, and generalizability across numerical experiments.
Takeaways & Limitations
Physics encoded in the architecture can support temporal extrapolation, generalization to different initial conditions, and extraction of parsimonious closed-form PDEs from imperfect measurements.
Takeaways & Limitations
The model is limited by standard convolutions on irregular meshes and by its assumption that governing PDEs have a polynomial form.

Abstract

from arXiv · show

Modeling complex spatiotemporal dynamical systems, such as the reaction-diffusion processes, have largely relied on partial differential equations (PDEs). However, due to insufficient prior knowledge on some under-explored dynamical systems, such as those in chemistry, biology, geology, physics and ecology, and the lack of explicit PDE formulation used for describing the nonlinear process of the system variables, to predict the evolution of such a system remains a challenging task. Unifying measurement data and our limited prior physics knowledge via machine learning provides us with a new path to solving this problem. Existing physics-informed learning paradigms impose physics laws through soft penalty constraints, whose solution quality largely depends on a trial-and-error proper setting of hyperparameters. Since the core of such methods is still rooted in black-box neural networks, the resulting model generally lacks interpretability and suffers from critical issues of extrapolation and generalization. To this end, we propose a deep learning framework that forcibly encodes given physics structure to facilitate the learning of the spatiotemporal dynamics in sparse data regimes. We show how the proposed approach can be applied to a variety of problems regarding the PDE system, including forward and inverse analysis, data-driven modeling, and discovery of PDEs. The resultant learning paradigm that encodes physics shows high accuracy, robustness, interpretability and generalizability demonstrated via extensive numerical experiments.

1 INTRODUCTION

Reaction-diffusion and other spatiotemporal systems are difficult to model because their governing PDEs may be uncertain, nonlinear, or computationally expensive to simulate. The proposed physics-encoded architecture hard-codes known physics into a discrete recurrent-convolutional network rather than imposing it only through penalized loss terms.

Motivation: Reaction-diffusion systems arise across chemistry, biology, geology, physics, and ecology, but governing equations for many underexplored systems remain uncertain or incomplete.The paper uses Turing-pattern formation as an example of dynamics governed by reaction and diffusion.
Motivation: Nonlinear reaction mechanisms and costly large-scale simulations make data-driven modeling attractive for complex spatiotemporal systems.Machine learning is presented as an avenue for scientific modeling and discovery when explicit equations or efficient simulation are unavailable.
Related work: Deep learning has enabled data-driven scientific computing, including complex-system modeling, super-resolution, material prediction, system identification, and equation discovery.The paper attributes these applications largely to deep learning's expressive power.
Related work: Physics-informed neural networks have been applied to PDE solving, reduced-order modeling, uncertainty quantification, inverse problems, and physical-system modeling.Examples include fluid dynamics, subsurface transport, and engineering mechanics.
Contribution: The proposed physics-encoded model embeds prior physics knowledge in the network architecture instead of teaching it through a penalized loss function.This design is introduced as the paper's central response to existing physics-informed learning approaches.
Contribution: Unlike continuous FCNN-based approximators, the model uses a discrete mesh-based solution, a convolutional Π-block for spatial patterns, recurrent time marching, and embedded numerical integration schemes.The architecture can encode incomplete PDEs using methods such as forward Euler and Runge-Kutta schemes.

RESULTS

PeRCNN encodes PDE structure directly into a recurrent convolutional architecture, enabling accurate modeling, extrapolation, generalization, inference, and PDE discovery from sparse noisy data. Across reaction-diffusion experiments, it remains interpretable because its learned expressions can be extracted and compared with governing equations.

Physics-encoded Spatiotemporal Learning: PeRCNN preserves known PDE structure, initial conditions, and boundary conditions in its architecture for robust prediction from limited low-resolution noisy measurements.The model is designed to learn spatiotemporal dynamics while forcibly encoding prior physics knowledge.
Forward Analysis: PeRCNN agrees well with reference solutions across four 2D and 3D reaction-diffusion cases, while ConvLSTM and PINN deviate more substantially in 3D.The comparison covers Lambda–Omega, FitzHugh-Nagumo, and Gray-Scott systems.
Inverse Analysis of PDE Systems: MARE is 0.6% without noise and 1.61% with 10% Gaussian noise for identified coefficients, demonstrating accurate inverse analysis under measurement noise.The experiments used multiple random-seed runs and also reported superiority to PINN.
Data-driven Modeling of Spatiotemporal Dynamics: PeRCNN outperforms competing data-driven models in long-time extrapolation and generalizes to unseen initial conditions, whereas several baselines produce inaccurate or wild predictions.The model is reported as the only method working well for long-time extrapolation in the cited 2D Gray-Scott example and as consistent with reference solutions for different initial conditions.
Data-driven Modeling of Spatiotemporal Dynamics: PeRCNN uses fewer trainable parameters while achieving higher accuracy, with training time per epoch comparable to ResNet in the 3D case.The reported efficiency comparison includes parameter counts, training time, and training and extrapolation RMSE.
Data-driven Discovery of PDEs with Scarce and Noisy Data: Integrated sparse regression enables recovery of governing PDEs completely on clean or mildly noisy data and recovery of most terms at 10% noise.Symbolic extraction from the multiplicative Π-block also yields explicit learned expressions that match genuine PDEs apart from minor near-zero terms or noise-related distractions.

DISCUSSION

PeRCNN encodes prior physics directly into its architecture, supporting interpretable PDE discovery and accurate modeling from sparse, noisy data. Its main limitations are computational cost, regular-grid dependence, and reliance on polynomial PDE structure.

PeRCNN hard-encodes initial and boundary conditions, PDE structure, and known PDE terms into its network architecture.This architecture is intended to improve convergence and accuracy while preserving the specified physics.
Extensive experiments show accurate forward and inverse analysis for reaction-diffusion systems, with extrapolation, generalizability, and robustness to sparse or noisy data.
PeRCNN supports interpretable PDE discovery by combining symbolic extraction from the Π-block with sparse regression.The coupled procedure iteratively optimizes network parameters and refines discovered PDE structures and coefficients.
High-dimensional discretization creates a computational bottleneck for large 3D domains with long-term evolution.The authors suggest temporal batching and multi-GPU training as possible remedies.
Standard convolution limits PeRCNN on irregular meshes, while its polynomial-PDE assumption may reduce suitability for systems using operators such as division, sin, exp, or log.

METHODS

The method builds a recurrent convolutional model from numerical discretization, embedding known physics through architecture rather than loss penalties. Its Π-block approximates nonlinear dynamics, while symbolic computation and sparse regression enable PDE discovery.

Physics-encoded spatiotemporal learning: PeRCNN uses a forward Euler update to recurrently advance the discretized state through an approximated PDE right-hand side.The update is pU^(k+1) = pU^k + pF(pU^k; θ)δt.
PeRCNN architecture: The architecture combines a fully convolutional initial-state generator with a recurrent Π-block for time evolution.The initial-state generator produces a high-resolution initial state from low-resolution measurements.
Π-block: The Π-block multiplies feature maps from parallel convolutional layers and combines channels to approximate nonlinear functions of the state and its spatial derivatives.
Physics encoding: Physics-based finite-difference convolution connections encode known PDE terms using fixed stencils, while physics-based padding encodes boundary conditions.This architecture can use incomplete PDE information without adding a separate physics penalty.
Approximation properties: The Π-block is designed as a universal polynomial approximator, and its approximation quality depends on the number of represented polynomial terms.Smaller polynomial representations can increase truncation error, while polynomial-form dynamics require fewer parameters for satisfactory accuracy.
PDE discovery: PDE discovery reconstructs high-fidelity solutions, extracts candidate terms symbolically from the Π-block, and applies sparse regression to identify governing equations.The candidate library is learned from the network rather than predefined conventionally.
PDE discovery: Coefficient fine-tuning uses the original measurements with trainable term coefficients to improve the accuracy of discovered PDEs.

A.1 Methodology for modeling scientific problems

Scientific modeling methods can be organized by their use of measurement data and physics knowledge. PeRCNN belongs to a physics-guided data-driven approach that uses numerical discretization and architecture design for dynamical-system modeling and discovery.

Physics-based, data-driven, and hybrid physics-guided data-driven models form three broad categories of scientific computational methods.
Figure S.1 classifies computational methods by measurement-data usage on the x-axis and physics or domain-knowledge usage on the y-axis.
Physics-guided data-driven models combine measurement data with partial physics knowledge and can reduce the need for large datasets.
Physics-informed learning: PeRCNN differs from soft physics-informed learning by encoding physics into a discrete, mesh-based network architecture rather than only informing training through penalties.
Relation to PDE-Net: The Π-block uses a multiplicative network form to represent power functions such as uΔu and u^2v that PDE-Net cannot represent readily.
Residual learning for dynamical systems: The model discretizes states on Cartesian grids and updates them recurrently using a forward Euler scheme with an approximated right-hand side.
Scientific modeling tasks: The paper targets PDE solving, high-fidelity prediction from scarce noisy measurements, and discovery of governing PDE forms or coefficients.

B.2 Design philosophy of the network architecture

PeRCNN combines a convolutional initial-state generator with a recurrent Π-block that updates discretized state variables while hard-encoding available physics. The architecture represents nonlinear dynamics through elementwise products of parallel convolutional features and can incorporate boundary conditions and known PDE terms.

Network components: PeRCNN uses a fully convolutional decoder to generate the high-resolution initial state and a recurrent Π-block to update state variables recursively.The model starts from low-resolution measurements and produces full-resolution predictions over discrete spatial grids and time steps.
Π-block design: The Π-block sends the previous state through parallel convolutional layers, multiplies their feature maps elementwise, and aggregates them with a 1 × 1 convolution.The aggregated output is trained to approximate the nonlinear right-hand side of the governing PDE.
Nonlinear representation: The architecture omits traditional nonlinear activation functions to improve interpretability when learning unknown physics.The design instead obtains nonlinearity through the product of parallel convolutional outputs.
Nonlinear representation: Parallel convolutional products provide a flexible representation of polynomial nonlinearities without explicitly specifying the basis functions or highest polynomial degree.A two-layer Π-block can represent polynomial terms such as u, ∆u, uv, and u ¨ ∇u.
Scope: The primary nonlinear function considered in the architecture is polynomial, while trigonometric and exponential forms are left for future extensions or polynomial approximation.The paper tests the Π-block on a reaction-diffusion equation with a non-polynomial reaction term.
Physics encoding: Initial and boundary conditions are encoded directly through physics-based padding, including periodic padding and extensions for Dirichlet, Neumann, and Robin conditions.Known PDE terms can also be inserted through highway convolutional layers with predefined finite-difference filters, while encoding prior physics narrows feasible model parameters.

C.2 2D problems

The 2D experiments evaluate PeRCNN on Lambda–Omega and FitzHugh–Nagumo reaction-diffusion systems. The reported results show accurate long-horizon prediction, including small error for the Lambda–Omega solution and satisfactory agreement for the slower FitzHugh–Nagumo dynamics.

Test systems: PeRCNN is evaluated on the 2D Lambda–Omega and FitzHugh–Nagumo reaction-diffusion systems.The Lambda–Omega case uses a spiral-pattern initial condition, while the FitzHugh–Nagumo case is designed to test slower evolution.
Lambda–Omega: 10^-7 maximum absolute error is reported for the 2D Lambda–Omega prediction against the reference solution.The spiral pattern is well predicted, and the error is small relative to the ground-truth scale.
FitzHugh–Nagumo: The FitzHugh–Nagumo experiment uses a 2D domain with 6001 predicted time steps and a Gaussian-distributed initial condition.The system parameters include µu = 1.0, µv = 10.0, α = 0.01, and β = 0.25.
Lambda–Omega: PeRCNN maintains satisfactory prediction despite the very large number of simulated time steps in the Lambda–Omega example.The authors use this example to demonstrate scalability to problems with long time patterns.

C.3 3D problems

The 3D experiments test PeRCNN on FitzHugh–Nagumo and Gray–Scott reaction-diffusion systems. The reported FitzHugh–Nagumo errors are small, and the experiments demonstrate application to high-dimensional 3D domains.

Scope: PeRCNN is tested on 3D FitzHugh–Nagumo and Gray–Scott reaction-diffusion systems to assess scalability beyond 2D problems.The paper frames 3D PDE solving as more challenging because of the curse of dimensionality.
Comparison: The authors report that PeRCNN solves the 3D reaction-diffusion examples while ConvLSTM and PINN struggle on these high-dimensional problems.This comparison is stated as a qualitative demonstration of the proposed architecture’s 3D capability.
Gray–Scott: The 3D Gray–Scott experiment covers a temporal domain from 0 to 750 with three randomly placed initial disturbances.The prediction uses 1501 time steps and is pretrained sequentially on 101, 251, and 501 time steps before full training.

C.4 Comparison with existing methods

PeRCNN is compared with PINN and ConvLSTM on reaction-diffusion prediction and is also adapted for coefficient identification from low-resolution, noisy measurements. It reports smaller prediction errors than the baselines and uses a completed physics-based recurrent block when the PDE form is known except for scalar coefficients.

Prediction comparison: The comparison evaluates PeRCNN, PINN, and ConvLSTM on 2D Lambda–Omega and FitzHugh–Nagumo reaction-diffusion systems.Hyperparameters for each method are selected from a range using a hold-out validation set.
Prediction comparison: PeRCNN has a much smaller error scale than the other baselines in the visual comparisons.PINN shows large boundary errors, whereas PeRCNN and ConvLSTM impose periodic boundary conditions through periodic padding.
Prediction comparison: Adding a highway diffusion convolution improves ConvLSTM accuracy, but an accuracy gap remains between ConvLSTM and PeRCNN.The modified ConvLSTM can solve only the 2D problems because of increased computational overhead.
Coefficient identification: Coefficient identification uses low-resolution, noisy snapshots of a 2D Gray–Scott system whose PDE form is known except for scalar coefficients.The unknown coefficients include diffusion, reaction, feed, and kill-related terms.
Coefficient identification: The coefficient-identification model replaces the Π-block with a completed physics-based recurrent network built from the known right-hand side of the PDE.This design switches between the Π-block and a fully physics-based block according to the availability of prior governing-equation knowledge.
Coefficient identification: The identification data comprise low-resolution snapshots with 10% uncorrelated Gaussian noise, and early stopping is used to reduce overfitting to noise.The measurements include 26 snapshots and are downsampled fourfold in both spatial directions.

D.3 Scenario 2: with initial and final high-resolution snapshots

This scenario tests whether PeRCNN can identify unknown coefficients from only initial and final high-resolution snapshots, including under noisy measurements. The method also extends to space-varying coefficients and data-driven high-resolution modeling.

Experimental scenario: Two snapshots at t = 450 and 650, each with 51 × 51 resolution, formed the training dataset for the 2D Gray-Scott system.The snapshots were corrupted with 10% Gaussian noise.
Coefficient identification: 10% noise produced a 1.05% MARE for identified coefficients, compared with 0.96% in the noise-free case.The network identified the unknown scalar coefficients accurately in both settings.
Space-varying coefficients: PeRCNN extends to space-varying coefficients by introducing a trainable approximator, implemented here with a five-component Gaussian mixture.The Gaussian-mixture parameters are optimized jointly with the network.
Data-driven modeling: The model targets reliable high-resolution full-field predictions from limited, noisy, coarse measurements while preserving temporal extrapolation capability.The modeling setup uses low-resolution measurements and learns the system dynamics through PeRCNN.
Architecture: PeRCNN architecture tests vary filter size, channel count, and the number of convolutional layers to control derivative approximation and nonlinear-term capacity.Filter size relates to the highest approximable spatial-derivative order, while channels determine how many nonlinear terms the Π-block can express.

E.5.1 Baselines

The baselines include recurrent, convolutional, differential-operator, and fully connected models evaluated against PeRCNN. Across extrapolation tests, PeRCNN maintains accuracy better than the alternatives and uses few parameters in the 3D case.

Baselines: Recurrent ResNet, ConvLSTM, PDE-Net, and DHPM provide diverse baselines spanning shared temporal weights, gated convolutions, learned differential operators, and hidden FCNN physics.The comparison uses configurations selected by validation loss from model-specific hyperparameter ranges.
Temporal extrapolation: All models except PeRCNN deteriorate sharply during temporal extrapolation, while PeRCNN outperforms competitors at every stage in accumulative RMSE.The training measurements use 26 × 26 resolution and 10% Gaussian noise; the initial high RMSE reflects noise in the first snapshot.
3D scalability: In the 3D Gray-Scott system, PeRCNN uses the fewest trainable parameters and has per-epoch training time comparable to ResNet.The experiment specifically examines scalability and computational efficiency in a higher-dimensional setting.
Interpretability: The Π-block exposes an equivalent explicit expression of the learned dynamics, improving interpretability over traditional fully connected or convolutional approaches.The learned diffusion coefficients are also identified, although noise and model redundancy introduce distracting terms.
Initial-condition generalization: PeRCNN generalizes to different initial conditions with predictions consistent with the ground truth, whereas recurrent ResNet and ConvLSTM produce wild predictions.DHPM is excluded from this comparison because it cannot perform inference with different initial conditions.

F.2 Methodology

The equation-discovery framework combines high-resolution data reconstruction, sparse regression, and coefficient fine-tuning. Applied to reaction-diffusion systems, it recovers active governing-equation terms and refines their coefficients.

F.2 Methodology: The framework uses three steps: reconstruct high-fidelity data, perform sparse regression, and fine-tune the resulting coefficients.The final recurrent network is physics-based after the explicit PDE structure is obtained.
Data reconstruction: PeRCNN reconstructs spatiotemporally high-resolution data before regression, reducing truncation errors caused by applying finite differences directly to noisy, low-resolution measurements.The 2D Gray-Scott example uses a Π-block with three parallel convolutional layers, eight channels, and filter size one.
Sparse regression: Sequential Threshold Ridge regression identifies active terms separately for each component of the 2D Gray-Scott system.The identified sets are S_u = {Δu, uv^2, u, 1u} and S_v = {Δv, uv^2, v u}.
Coefficient fine-tuning: Fine-tuning uses the explicit sparse-regression PDE in a physics-based recurrent network to improve coefficient accuracy using the available measurements.The regression initially uses subsampled reconstructed high-resolution data, motivating the subsequent refinement step.

F.3 Experimental setup

The experiments use synthetic, downsampled, noise-corrupted reaction-diffusion and Burgers’ data to evaluate equation discovery. Performance is assessed by coefficient error and by precision and recall of recovered terms.

Experimental setup: The evaluation covers 2D λ–Ω, 2D Gray-Scott, and 2D Burgers’ systems using synthetic numerical examples.The governing equations and simulation parameters are provided for the systems used in the study.
Sparse regression algorithm: Sequential Threshold Ridge regression initializes with ridge regression, thresholds small coefficients to zero, updates the remaining coefficients, and finalizes them by least squares.The procedure stops after the iteration limit or when the set of large coefficients remains unchanged for two consecutive iterations.
Evaluation metrics: Relative ℓ2 error measures coefficient-vector distance, while precision and recall evaluate correct term selection from the candidate library.Precision counts identified nonzero terms that are truly nonzero; recall measures the fraction of true nonzero terms recovered.
Visual evaluation: Figure S.17 compares low-resolution measurements, high-resolution truth, reconstructed PeRCNN solutions, PDE-Net, FCNN, and polynomial fitting, with reconstruction errors reported.The comparison spans 2D Burgers’, λ–Ω, and Gray-Scott systems.

F.4 Results and comparison with existing methods

Across reaction-diffusion and Burgers’ examples, PeRCNN reconstructs high-resolution dynamics and supports PDE discovery from sparse or noisy data. It generally outperforms or remains more robust than baseline approaches, while its discovered equations can depend on sparse-regression choices.

Under 10% noise, the Gray-Scott discovery reaches 5.9×10^-3 relative ℓ2 error with 100% recall and precision, while baseline performance deteriorates.
Under 10% noise, the method achieves 5.44×10^-2 relative error, 100% recall, and 91.6% precision for discovered PDEs.
For the 2D Gray-Scott system, PeRCNN reconstructs high-resolution data from 26×26 measurements and discovers PDE terms that competing methods struggle to identify.The comparison cites PDE-FIND, PDE-Net+SR, and FCNN+SR as baselines affected by extremely low-resolution measurements.
The learned Π-block can supply an analytical expression for constructing the sparse-regression library, but predefined candidate sets remain necessary when extraction is unavailable.
Fine-tuning improves coefficient accuracy, but the final discovered PDE depends substantially on whether sparse regression initially identifies the correct PDE form.

F.5.5 Scalability to the noise level and data sparsity

Noise tolerance depends on spatial resolution and pattern complexity, while discoverability also depends on data sparsity. The method remains competitive for smoother, better-resolved systems but degrades sharply for low-resolution systems with localized structure.

For 2D Gray-Scott, accuracy deteriorates drastically as noise increases because 26×26 measurements blur local patterns and sharp gradients.
At 30% Gaussian noise, the method remains competitive for 2D λ–ΩRD and Burgers’ equations with 51×51 measurements and smoother spatial patterns.
Overall, the experiments identify a trade-off between measurement sparsity and noise level.
The lowest resolution supporting Gray-Scott PDE discoverability is 26×26, whereas λ–ΩRD yields 90.9% precision and 83.3% recall at 11×11 resolution.

G An example with the Neumann BC type

PeRCNN encodes Neumann boundary conditions through convolutional padding and accurately solves a 2D heat-transfer problem. Resolution choices are constrained by stability, convergence, and available computational resources.

Neumann boundaries are enforced by padding predicted snapshots with ghost nodes so the central-difference normal derivative equals zero.
PeRCNN’s heat-transfer prediction agrees very well with the finite-difference reference across 1000 high-resolution 101×101 snapshots.
Spatial and temporal steps are selected to satisfy stability and convergence requirements, including µδt/δx^2 ≤ 1/2.
Finer resolutions improve spatiotemporal detail but are limited by computational resources; parallel computing is proposed for scaling.

I Data-driven modeling of a PDE with non-polynomial term

The framework is tested on a reaction-diffusion PDE containing non-polynomial terms. With noisy, lower-resolution measurements, it predicts high-resolution dynamics and extrapolates beyond the training interval with good agreement to the ground truth.

The experiment evaluates PeRCNN on a variant of the 2D Gray-Scott system containing non-polynomial terms.
Predicted snapshots show good agreement with the ground truth, including inference beyond the training time interval.

J Extension of PeRCNN for modeling turbulent flows

PeRCNN is extended to model turbulent flows by evolving vorticity while recovering velocity through known Poisson equations. On decaying turbulence at Re = 1,000, the model generalizes to a different initial condition and unrolls accurate predictions beyond its training horizon.

Model formulation: The extended PeRCNN predicts velocity and vorticity fields by temporally unrolling vorticity and updating velocity through Poisson equations solved with FFT.The Poisson equations provide prior physics knowledge for recovering the velocity components at each time step.
Model formulation: Two parallel convolutional layers with 5 ˆ 5 filters and 32 channels incorporate additional prior physics into the extended PeRCNN design.The authors report preliminary improvements in convergence and approximation ability from incorporating additional physics such as the continuity equation.
Problem setup: The turbulent-flow setup uses Re = 1,000 in a periodic domain, with DNS reference data generated on a 1,024ˆ1,024 grid.The DNS time step is δt = 3.6523 ˆ 10´4 and satisfies the CFL condition.
Generalization results: Figure S.24 compares predicted vorticity and velocity evolutions with their correlations against the reference solution over time.The reference solution is down-sampled from the 1,024ˆ1,024 DNS grid to 128ˆ128 resolution, with each displayed step representing ∆t = 32δt.
Generalization results: Despite training on only 480 steps, the trained model predicts a different-initial-condition turbulent flow for 1,000 steps with satisfactory accuracy.Predicted flow patterns match the ground truth, while accumulated error remains within a reasonable range.