Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics

TL;DR

This work introduces the PDE-preserved neural network (PPNN), a multi-resolution, auto-regressive framework that bakes discretized PDE operators directly into convolutional residual blocks to predict spatiotemporal dynamics. By preserving part or all of the governing PDEs on coarse grids and coupling them with trainable high-resolution components, PPNN delivers strong long-horizon accuracy and robust generalization across a range of PDEs, including FitzHugh–Nagumo RD, Burgers’, and Navier–Stokes, even when physics are partially known or incomplete. Compared with state-of-the-art neural operators and physics-informed baselines (PINN, DeepONet, FNO), PPNN achieves lower predictive errors, reduced error growth in time, and substantial speedups in inference, making it well-suited for repeated multi-query tasks such as optimization, data assimilation, and uncertainty quantification. The framework is demonstrated across fully known, partially known, and even mis-specified PDEs, showing both the value and limitations of embedding physical priors into neural architectures and highlighting directions for extending to unstructured data and uncertainty quantification.

Abstract

Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by incorporating physical principles into the model. Most PiDL approaches regularize training by embedding governing equations into the loss function, yet this depends heavily on extensive hyperparameter tuning to weigh each loss term. To this end, we propose to leverage physics prior knowledge by ``baking'' the discretized governing equations into the neural network architecture via the connection between the partial differential equations (PDE) operators and network structures, resulting in a PDE-preserved neural network (PPNN). This method, embedding discretized PDEs through convolutional residual networks in a multi-resolution setting, largely improves the generalizability and long-term prediction accuracy, outperforming conventional black-box models. The effectiveness and merit of the proposed methods have been demonstrated across various spatiotemporal dynamical systems governed by spatiotemporal PDEs, including reaction-diffusion, Burgers', and Navier-Stokes equations.

Figures (27)

• Figure 1: Schematic diagram of the proposed partial differential equation (PDE)-preserved neural network (PPNN). a. A schematic representation illustrating the concept of the PPNN framework. b. A detailed schematic of the ConvResNet-based PPNN, which consists of the trainable part and the PDE-preserving part. The two portions of PPNN are combined together in a multi-resolution setting. The discretized form of the governing PDEs are embedded into the network structure via prescribed convolutions filters and the residual connection.
• Figure 2: Prediction comparison in the reaction-diffusion (RD) case and viscous Burgers' case. a and b, Predicted solution snapshots of $u$ for the RD equations (a), and the velocity magnitude $\lVert\bm{u}\rVert_2$ for the Burgers' equations (b) at different time steps and unseen parameters, obtained by black-box ConvResNet (baseline model, first rows), and partial differential equation preserved neural network (PPNN, our method, second rows), compared against ground truth (high-resolution numerical simulation, third rows). ${\lambda}_0$, ${\lambda}_1$ are randomly selected testing (unseen) parameters in each system. c - f, Relative prediction error $\epsilon_t$ of PPNN (blue lines ) and black-box ConvResNet baseline (orange lines ) for the RD dynamics (c-d) and Burgers' equations (e - f), averaged on 100 randomly sampled training parameters $\bm{\lambda}$ (c, e) and testing (unseen) parameters (d, f). The shaded area shows the maximum and minimum relative errors of all testing trajectories. g, h, Zoom in views of the relative error curve of PPNN shown in c (g) and d (h), respectively. i, j, Zoom in views of the relative error curve of PPNN shown in e (i) and f (j), respectively.
• Figure 3: Prediction comparison in the case governed by Naiver-Stocks (NS) equations. a-b, Predicted solution snapshots of velocity magnitude $\lVert \bm{u}\rVert_2$ for the NS equations obtained by black-box ConvResNet (baseline); partial differential equation preserved neural network (PPNN, Ours), compared against the ground truth (high-resolution numerical simulation), where $\bm{\lambda}_0$ is ($\mathrm{Re} = 2500, y_0 = 0.325$, shown in a), and $\bm{\lambda}_1$ is high Reynolds number ($\mathrm{Re} = 8500, y_0 = 0.575$, shown in b). c-d, Relative prediction error $\epsilon_t$ of PPNN (blue lines ) and black-box ConvResNet baseline (orange lines ) at different timesteps for the NS equation, averaged on 5 randomly sampled (c) training parameters and (d) testing (unseen) parameters. The shaded areas show the scattering of the relative errors over all testing trajectories.
• Figure 4: Prediction comparison in the cases where the governing equations are partially known. a, Predicted solution snapshots of $u$ for the reaction-diffusion (RD) equations at different time steps and unseen parameters, obtained by black-box ConvResNet (baseline model), and partial differential equation (PDE)-preserved neural network (PPNN, preserving diffusion terms only), and PPNN (preserving complete FitzHugh–Nagumo RD equations), compared against ground truth. ${\lambda}_2$ and ${\lambda}_3$ are two randomly selected testing (unseen) parameters. b-c, Averaged relative testing error $\epsilon_t$ of the PPNN with incomplete PDE (blue lines ) and Black-Box ConvResNet baseline (orange lines ) for the RD dynamics evaluated on 100 randomly generated testing (unseen) parameters (same parameters as shown in Fig. \ref{['fig:rd_bg']}c). Shaded areas in b indicate envelopes of the maximum and minimum relative errors of all testing trajectories, while the dash lines in c indicate the relative error of each test trajectory. d-e, Predicted solution snapshots of flow velocity magnitude $\lVert \bm{u}\rVert_2$ obtained by black-box ConvResNet (baseline), PPNN (ours), compared against ground truth (high-resolution numerical simulation) of the NS equations without (d) and with (e) an unknown magnetic source term, respectively. The PPNN only preserves a NS equation portion for both scenarios, which are at the same testing (unseen) parameter $\bm{\lambda} = [9000, 0.475]^T$, which is not in the training set. f-g, Relative prediction errors $\epsilon_t$ of the PPNN (blue line ) and black-box ConvResNet baseline (orange line ) for the NS equation with (e) and without (f) a unknown magnetic body force, averaged on five randomly sampled unseen parameters. The shaded area shows the scattering of relative errors for all testing trajectories.
• Figure 5: Relative error $\epsilon_t$ comparison when wrong terms are embedded in partial differential equation preserved neural network (PPNN), tested on 2D Burgers' equation. The relative error of PPNN (blue line ), its black-box counterpart (orange line), and PPNN with completely wrong partial differential equation terms (green line ) tested on unseen parameters is shown in the figure. Solid lines show the mean relative error, while the shaded areas show the distribution of all the 100 sample trajectories.