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

TL;DR

MultiPDENet presents a PDE-embedded learning framework with multi-scale time stepping to accelerate flow simulations on coarse grids while preserving physical fidelity. It combines a Learnable Physics Block (including a PDE Block and a Poisson Block with RK4 integration) and correction NN modules (M_iNn and M_aNn) with a symmetry-constrained adaptive filter to approximate spatial derivatives on coarse grids. The approach enables long-term, accurate predictions across diverse PDE systems (KdV, Burgers, Gray-Scott, NSE) and demonstrates strong generalization to unseen ICs, forcing terms, Reynolds numbers, and domain sizes, achieving state-of-the-art accuracy and notable speedups over traditional DNS. Key contributions include the RK4-based learnable solver, the adaptive symmetric filter for derivative estimation, and the multi-scale macro/micro stepping strategy that mitigates temporal error accumulation. Overall, MultiPDENet offers a robust, efficient path to fast, generalizable flow simulations with limited training data, suitable for applications requiring rapid scenario testing and design optimization.

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 (10)

• Figure 1: Schematic of MultiPDENet for learning turbulent flows. (a), Model architecture. (b), Physics Block. (c), Learnable PDE block.
• Figure 2: Symmetric filter
• 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$\times$25, 32$\times$32, respectively.
• 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 steps. (c) shows the scaled energy spectrum scaled by $k^5$ averaged between time steps 100 and 500.
• 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). (a-c) display results for different Reynolds numbers, (d-f) show results for varying external forcings.