Long-time Integration of Nonlinear Wave Equations with Neural Operators

TL;DR

This work tackles the challenge of long-time prediction for nonlinear wave equations using neural operators by reframing how predictions are rolled out over time. It introduces window-type input strategies, recurrent versus full-window architectures, randomizing the initial time, and regularization based on conservation laws and Strichartz-inspired clipping, primarily within Fourier Neural Operator (FNO) frameworks. The approach is validated on KdV, sine-Gordon, and Klein-Gordon equations, showing improved stability, better handling of nonlinear phenomena, and effective operation on irregular geometries via Geo-FNO. The methods offer a practical path to accurate, meshless long-time simulations with potential extensions to physics-informed and PDE-aware models.

Abstract

Neural operators have shown promise in solving many types of Partial Differential Equations (PDEs). They are significantly faster compared to traditional numerical solvers once they have been trained with a certain amount of observed data. However, their numerical performance in solving time-dependent PDEs, particularly in long-time prediction of dynamic systems, still needs improvement. In this paper, we focus on solving the long-time integration of nonlinear wave equations via neural operators by replacing the initial condition with the prediction in a recurrent manner. Given limited observed temporal trajectory data, we utilize some intrinsic features of these nonlinear wave equations, such as conservation laws and well-posedness, to improve the algorithm design and reduce accumulated error. Our numerical experiments examine these improvements in the Korteweg-de Vries (KdV) equation, the sine-Gordon equation, and the Klein-Gordon wave equation on the irregular domain.

Figures (10)

• Figure 1: The recurrent network and full prediction network. The operator to be learned exhibits a window-to-window mapping structure. The full prediction network (red box) learns this mapping directly and predicts the full output window in one step. The recurrent network (grey box) uses the neural operator to predict a snapshot of the next moment in each step. The snapshot is then appended to the shifting input window and the first moment snapshot is discarded. This process continues until the next window is completely predicted.
• Figure 2: Accumulation error for KdV equation. (a): Comparison between DeepONet and FNO. (b)(c): Comparison between FNO trained with and without conservation law regularization. (d)(e): Comparison between FNO trained with vanilla data and random data.
• Figure 3: Predictions for KdV equation. (a): Prediction generated by the full prediction FNO trained with $\lambda = 10$. (b): Prediction generated by the full prediction FNO trained with local random data. The initial $10$ snapshots have zero error as they are the input window.
• Figure 4: Accumulation error for sine-Gordon equation. (a): Comparison between recurrent and full prediction FNO trained with vanilla data and random data. The grey area shows the trained range and interpolation error of each model. (b): Comparison between recurrent FNO trained with different window sizess $l$.
• Figure 5: Prediction for sine-Gordon equation. The prediction is made by the recurrent FNO trained with global random initial data.