提出新框架让神经算子严格遵守守恒定律,提升物理模拟准确性。
An Exterior-Embedding Neural Operator Framework for Preserving Conservation Laws
- 通过编码器-解码器结构提取并修正守恒量,强制模型输出满足守恒律。
- 在绝热系统、浅水方程等任务中,精度显著提升且守恒性严格满足。
- 通用框架可适配多种神经算子,适合需要高物理保真的仿真场景。
神经算子在加速求解时变偏微分方程(PDEs)方面表现出色,可通过数据直接学习物理规律。然而,对于受守恒律(如质量、能量或物质守恒)约束的PDE,现有神经算子无法保证守恒性质,导致性能下降和泛化能力受限。我们发现,不同PDE问题通常需要不同的最优网络架构,这凸显了专用模型在跨领域泛化上的内在局限性。为此,我们提出外嵌式守恒框架(ECF),一种可与多种数据驱动神经算子集成的通用守恒框架,能严格确保预测结果满足守恒律。该框架包含两个关键组件:守恒量编码器,从输入数据中提取守恒量;守恒量解码器,利用这些量调整神经算子的预测,以确保最终输出严格符合守恒要求。由于架构强制满足守恒律,我们理论上证明其能提升模型性能。我们在多个受守恒律约束的PDE场景中进行实验,包括绝热系统、浅水方程和Allen-Cahn问题,结果表明该方法在保持严格守恒的同时显著提升了模型精度。
原文摘要 · Abstract (English)
Neural operators have demonstrated considerable effectiveness in accelerating the solution of time-dependent partial differential equations (PDEs) by directly learning governing physical laws from data. However, for PDEs governed by conservation laws(e.g., conservation of mass, energy, or matter), existing neural operators fail to satisfy conservation properties, which leads to degraded model performance and limited generalizability. Moreover, we observe that distinct PDE problems generally require different optimal neural network architectures. This finding underscores the inherent limitations of specialized models in generalizing across diverse problem domains. To address these limitations, we propose Exterior-Embedded Conservation Framework (ECF), a universal conserving framework that can be integrated with various data-driven neural operators to enforce conservation laws strictly in predictions. The framework consists of two key components: a conservation quantity encoder that extracts conserved quantities from input data, and a conservation quantity decoder that adjusts the neural operator's predictions using these quantities to ensure strict conservation compliance in the final output. Since our architecture enforces conservation laws, we theoretically prove that it enhances model performance. To validate the performance of our method, we conduct experiments on multiple conservation-law-constrained PDE scenarios, including adiabatic systems, shallow water equations, and the Allen-Cahn problem. These baselines demonstrate that our method effectively improves model accuracy while strictly enforcing conservation laws in the predictions.
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