用神经网络构建满足守恒律的偏微分方程求解器，提升长期预测稳定性。
Hamiltonian Neural PDE Solvers through Functional Approximation
- 将哈密顿泛函建模为神经场核积分，实现函数到标量的可学习映射。
- 在1D和2D PDE上保持能量等守恒量，比传统方法更稳定且泛化更强。
- 适合需要物理一致性建模的科学计算场景，如流体、波动系统模拟。
在哈密顿框架下设计神经网络能确保物理系统中的守恒律被满足。尽管前景广阔，现有方法主要局限于离散且解析可解的系统。而多数物理现象由偏微分方程（PDE）描述，其本质是通过哈密顿泛函及其泛函导数控制无穷维场。本文基于前期工作，将哈密顿泛函表示为由神经场参数化的核积分形式，实现可学习的函数到标量映射，并利用自动微分计算泛函导数。由此扩展哈密顿力学至神经PDE求解器，通过预测泛函并学习梯度域进行训练。结果表明，所提出的哈密顿神经求解器（HNS）在1D与2D PDE中可作为高效代理模型，显著提升稳定性并持续保持能量类守恒量。该守恒特性也使HNS在更长时间尺度或未见过的初值条件下具备更强泛化能力。
原文摘要 · Abstract (English)
Designing neural networks within a Hamiltonian framework offers a principled way to ensure that conservation laws are respected in physical systems. While promising, these capabilities have been largely limited to discrete, analytically solvable systems. In contrast, many physical phenomena are governed by PDEs, which govern infinite-dimensional fields through Hamiltonian functionals and their functional derivatives. Building on prior work, we represent the Hamiltonian functional as a kernel integral parameterized by a neural field, enabling learnable function-to-scalar mappings and the use of automatic differentiation to calculate functional derivatives. This allows for an extension of Hamiltonian mechanics to neural PDE solvers by predicting a functional and learning in the gradient domain. We show that the resulting Hamiltonian Neural Solver (HNS) can be an effective surrogate model through improved stability and conserving energy-like quantities across 1D and 2D PDEs. This ability to respect conservation laws also allows HNS models to better generalize to longer time horizons or unseen initial conditions.
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