Predicting Wave Dynamics using Deep Learning with Multistep Integration Inspired Attention and Physics-Based Loss Decomposition

TL;DR

MI2A introduces a physics-informed deep learning framework that unites classical multistep time integration with attention-based latent dynamics to predict wave propagation governed by hyperbolic PDEs. By operating in a reduced latent space created by a denoising convolutional autoencoder and by using an attention-enabled evolver that mirrors multistep time-stepping, the approach achieves improved long-horizon stability and accuracy. A key contribution is the dispersion–dissipation loss decomposition, which separately targets phase (dispersion) and amplitude (dissipation) errors to improve generalization across parameter regimes. The method demonstrates strong performance across 1D linear convection, 1D Burgers, and 2D Saint-Venant problems, offering substantial speed-ups and robust predictive capability for real-time wave modeling in fluid dynamics and geophysical applications.

Abstract

In this paper, we present a physics-based deep learning framework for data-driven prediction of wave propagation in fluid media. The proposed approach, termed Multistep Integration-Inspired Attention (MI2A), combines a denoising-based convolutional autoencoder for reduced latent representation with an attention-based recurrent neural network with long-short-term memory cells for time evolution of reduced coordinates. This proposed architecture draws inspiration from classical linear multistep methods to enhance stability and long-horizon accuracy in latent-time integration. Despite the efficiency of hybrid neural architectures in modeling wave dynamics, autoregressive predictions are often prone to accumulating phase and amplitude errors over time. To mitigate this issue within the MI2A framework, we introduce a novel loss decomposition strategy that explicitly separates the training loss function into distinct phase and amplitude components. We assess the performance of MI2A against two baseline reduced-order models trained with standard mean-squared error loss: a sequence-to-sequence recurrent neural network and a variant using Luong-style attention. To demonstrate the effectiveness of the MI2A model, we consider three benchmark wave propagation problems of increasing complexity, namely one-dimensional linear convection, the nonlinear viscous Burgers equation, and the two-dimensional Saint-Venant shallow water system. Our results demonstrate that the MI2A framework significantly improves the accuracy and stability of long-term predictions, accurately preserving wave amplitude and phase characteristics. Compared to the standard long-short term memory and attention-based models, MI2A-based deep learning exhibits superior generalization and temporal accuracy, making it a promising tool for real-time wave modeling.

Figures (14)

• Figure 1: Schematic representation of the encode-propagate-decode architecture, highlighting its key components and their interactions.
• Figure 2: A diagram depicting convolutional autoencoder architecture employed for the dimensionality reduction.
• Figure 3: Illustration of the proposed attention-based sequence-to-sequence evolver. While the encoder generates hidden state vectors $H$ by transforming input, the decoder generates hidden state ($S$) by iterating over final encoder hidden state $h^{(n)}$. Notably the alignment score between $H$ and $S$ are computed.
• Figure 4: Visualization of MI2A architecture. Three blocks are shown namely the convolution encoder for creating the latent low-dimensional representation, the evolver for propagating the low-dimensional feature in time and the decoder for transforming the low dimension space to input data space.
• Figure 5: Linear convection problem: Exact solution (left), MI2A solution with n = 2 (center) and error $e = |\hat{u} - u|$ (right) for the testing parameter $\mu_{test}$ = 0.7875 in the space-time domain.