用记忆机制提升时间依赖偏微分方程的求解精度，尤其在低分辨率和噪声环境下。
On the Benefits of Memory for Modeling Time-Dependent PDEs
- 引入记忆神经算子MemNO，融合S4与FNO，显式利用历史状态预测未来。
- 在低分辨率或含噪声条件下，测试误差降低达6倍，优于无记忆基线。
- 对高频模式显著的流体动力学问题效果突出，适合高精度科学计算场景。
数据驱动方法已成为求解偏微分方程（PDEs）的有前景替代方案。对于时间依赖的PDEs，许多方法采用马尔可夫性假设——系统演化仅依赖当前状态，忽略历史。本文研究使用记忆建模时间依赖PDEs的优势：即显式利用过去状态预测未来。受莫里-曾万格降维理论启发，我们从理论上证明了某些简单（甚至线性）的PDE中，带记忆的解可任意优于马尔可夫解。同时提出记忆神经算子（MemNO），结合最近的状态空间模型（S4）与傅里叶神经算子（FNO），有效建模记忆。实验表明，在低分辨率输入或训练/测试时存在观测噪声的情况下，MemNO显著优于无记忆基线，测试误差最高降低6倍。此外，该优势在具有显著高频傅里叶模式的PDE中尤为明显（如低粘性流体动力学），我们构建了一个包含此类PDE的挑战性基准数据集。
原文摘要 · Abstract (English)
Data-driven techniques have emerged as a promising alternative to traditional numerical methods for solving PDEs. For time-dependent PDEs, many approaches are Markovian -- the evolution of the trained system only depends on the current state, and not the past states. In this work, we investigate the benefits of using memory for modeling time-dependent PDEs: that is, when past states are explicitly used to predict the future. Motivated by the Mori-Zwanzig theory of model reduction, we theoretically exhibit examples of simple (even linear) PDEs, in which a solution that uses memory is arbitrarily better than a Markovian solution. Additionally, we introduce Memory Neural Operator (MemNO), a neural operator architecture that combines recent state space models (specifically, S4) and Fourier Neural Operators (FNOs) to effectively model memory. We empirically demonstrate that when the PDEs are supplied in low resolution or contain observation noise at train and test time, MemNO significantly outperforms the baselines without memory -- with up to 6x reduction in test error. Furthermore, we show that this benefit is particularly pronounced when the PDE solutions have significant high-frequency Fourier modes (e.g., low-viscosity fluid dynamics) and we construct a challenging benchmark dataset consisting of such PDEs.
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