广义反应-扩散系统的神经本构学习

中文摘要 AI 辅助

广义反应-扩散系统涵盖了多种输运机制和耦合反应动力学。神经偏微分方程求解器的一个核心问题是，应学习什么内容，以便一个通用接口能够适应相场和退化输运、局部反应以及多物种耦合。我们提出了神经本构定律-质量-压缩-输运（NCL-MCT）求解器，该求解器学习特定于偏微分方程的本构响应，同时在共享的MCT积分器中保留时间演化。输运通过迁移率和热力学驱动力来表示，反应通过相对反应速率来表示。这些本构响应依赖于当前密度，而非显式地依赖于初始条件或经过的时间，这促使它们可在不同的初始条件和时间范围内重复使用。同一接口支持速度数据监督和已知定律监督，两者在训练期间均不需要时间积分。当本构定律已知时，可以在独立采样的密度场上评估监督，从而实现无需生成解轨迹的无轨迹本构学习。在七个系统中，分别训练的本构模块共享相同的接口和MCT积分器，并实现了相对滚动$L^2$误差在$10^{-4}$到$10^{-2}$之间。对未见过的初始条件族和扩展时间范围的测试评估了超出训练条件的重用性，而单独的实验则证明了无轨迹本构学习。这些结果支持本构响应作为共享神经偏微分方程框架的有效学习目标。

英文摘要

Generalized reaction-diffusion systems encompass diverse transport mechanisms and coupled reaction kinetics. A central question for neural PDE solvers is what should be learned so that a common interface can accommodate phase-field and degenerate transport, local reactions, and multispecies coupling. We propose the Neural Constitutive Laws--Mass-Compression-Transport (NCL-MCT) Solver, which learns PDE-specific constitutive responses while retaining temporal evolution in a shared MCT integrator. Transport is represented through mobility and thermodynamic driving force, and reaction through relative reaction rates. These constitutive responses depend on the current density rather than explicitly on the initial condition or elapsed time, motivating their reuse across different initial conditions and time horizons. The same interface supports velocity-data supervision and known-law supervision, neither of which requires time integration during training. When constitutive laws are known, supervision can be evaluated on independently sampled density fields, enabling trajectory-free constitutive learning without generating solution trajectories. Across seven systems, separately trained constitutive modules share the same interface and MCT integrator and achieve relative rollout $L^2$ errors of $10^{-4}$ to $10^{-2}$. Tests with unseen initial-condition families and an extended time horizon assess reuse beyond training conditions, while separate experiments demonstrate trajectory-free constitutive learning. These results support constitutive responses as an effective learning target for a shared neural PDE framework.

发表机构

• National Tsing Hua University（国立清华大学）
• NVIDIA（英伟达）

机构由 AI 辅助整理，请以论文原文为准。