On the Benefits of Memory for Modeling Time-Dependent PDEs

• URL: http://arxiv.org/abs/2409.02313v1
• Date: Tue, 3 Sep 2024 21:56:13 GMT
• Title: On the Benefits of Memory for Modeling Time-Dependent PDEs
• Authors: Ricardo Buitrago Ruiz, Tanya Marwah, Albert Gu, Andrej Risteski,
• Abstract summary: We introduce Memory Neural Operator (MemNO), a network based on the recent SSM architectures and Fourier Neural Operator (FNO) MemNO significantly outperforms the baselines without memory, achieving more than 6 times less error on unseen PDEs.
• Score: 35.86010060677811
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Data-driven techniques have emerged as a promising alternative to traditional numerical methods for solving partial differential equations (PDEs). These techniques frequently offer a better trade-off between computational cost and accuracy for many PDE families of interest. For time-dependent PDEs, existing methodologies typically treat PDEs as Markovian systems, i.e., the evolution of the system only depends on the ``current state'', and not the past states. However, distortion of the input signals -- e.g., due to discretization or low-pass filtering -- can render the evolution of the distorted signals non-Markovian. In this work, motivated by the Mori-Zwanzig theory of model reduction, we investigate the impact of architectures with memory for modeling PDEs: that is, when past states are explicitly used to predict the future. We introduce Memory Neural Operator (MemNO), a network based on the recent SSM architectures and Fourier Neural Operator (FNO). We empirically demonstrate on a variety of PDE families of interest that when the input is given on a low-resolution grid, MemNO significantly outperforms the baselines without memory, achieving more than 6 times less error on unseen PDEs. Via a combination of theory and experiments, we show that the effect of memory is particularly significant when the solution of the PDE has high frequency Fourier components (e.g., low-viscosity fluid dynamics), and it also increases robustness to observation noise.

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