MINPO：基于记忆的神经伪算子以解决非局部时空动态
MINPO: Memory-Informed Neural Pseudo-Operator to Resolve Nonlocal Spatiotemporal Dynamics
- Energy \& Intelligence Lab, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah 84112, USA
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
MINPO通过神经伪算子统一建模非局部时空动态，有效解决由积分微分方程描述的复杂问题。
AI中文摘要：
许多物理系统表现出由积分微分方程（IDEs）描述的非局部时空行为。经典方法求解IDEs需要反复计算卷积积分，其成本随着核的复杂性和维度性迅速增加。现有的神经求解器可以加速某些实例的计算，但无法在多样化的非局部结构上泛化。在本文中，我们引入了记忆感知神经伪算子（MINPO），这是一个统一的框架，用于建模由长程空间相互作用和/或长期时间记忆引起的非局部动态。MINPO通过使用Kolmogorov-Arnold网络（KANs）或多层感知机网络（MLPs）作为编码器，直接通过神经表示学习非局部算子及其逆运算，并显式重建未知解场。学习过程由一个轻量级的非局部一致性损失项保护，以确保学习到的算子和重建的解之间的一致性。MINPO的公式允许自然捕捉并高效解决由广泛IDEs及其子集（包括分数PDEs）所支配的非局部时空依赖关系。我们评估了MINPO与经典技术以及基于MLPs的最新神经求解器（如A-PINN和fPINN）以及其新开发的KAN变种（A-PIKAN和fPIKAN）的效能，以促进公平比较。我们的研究提供了MINPO准确性的有力证据，并展示了其在处理（i）不同类型的核、（ii）不同维度的核以及（iii）由于重复计算核积分而产生的大量计算需求方面的鲁棒性。MINPO因此超越了问题特定的公式，提供了一个由非局部算子支配的系统的统一框架。
英文摘要：
Many physical systems exhibit nonlocal spatiotemporal behaviors described by integro-differential equations (IDEs). Classical methods for solving IDEs require repeatedly evaluating convolution integrals, whose cost increases quickly with kernel complexity and dimensionality. Existing neural solvers can accelerate selected instances of these computations, yet they do not generalize across diverse nonlocal structures. In this work, we introduce the Memory-Informed Neural Pseudo-Operator (MINPO), a unified framework for modeling nonlocal dynamics arising from long-range spatial interactions and/or long-term temporal memory. MINPO, employing either Kolmogorov-Arnold Networks (KANs) or multilayer perceptron networks (MLPs) as encoders, learns the nonlocal operator and its inverse directly through neural representations, and then explicitly reconstruct the unknown solution fields. The learning is guarded by a lightweight nonlocal consistency loss term to enforce coherence between the learned operator and reconstructed solution. The MINPO formulation allows to naturally capture and efficiently resolve nonlocal spatiotemporal dependencies governed by a wide spectrum of IDEs and their subsets, including fractional PDEs. We evaluate the efficacy of MINPO in comparison with classical techniques and state-of-the-art neural-based strategies based on MLPs, such as A-PINN and fPINN, along with their newly-developed KAN variants, A-PIKAN and fPIKAN, designed to facilitate a fair comparison. Our study offers compelling evidence of the accuracy of MINPO and demonstrates its robustness in handling (i) diverse kernel types, (ii) different kernel dimensionalities, and (iii) the substantial computational demands arising from repeated evaluations of kernel integrals. MINPO, thus, generalizes beyond problem-specific formulations, providing a unified framework for systems governed by nonlocal operators.
