发表机构
University of California, Los Angeles（加州大学洛杉矶分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究针对偏微分方程基础神经模拟器面临的条件无关部署、物理保真度和推理效率瓶颈，提出广义神经算子，通过形式化适定性条件，引入三个新颖架构组件，实现跨异构物理状态的卓越泛化与高效推理。
AI 中文摘要
开发用于偏微分方程（PDE）的基础神经模拟器需要在不同物理参数和边界条件下具有强大的泛化能力。当前深度学习方法在条件无关部署和物理保真度之间面临结构权衡。纯数据驱动算子隐式推断物理原理，缺乏确保跨不同域物理有效解所需的显式约束，使学习问题不适定。物理信息神经网络（PINNs）虽强制执行严格物理约束，但需要针对特定实例进行昂贵优化。此外，新兴基础算子的大规模计算严重降低了推理速度。为解决条件无关部署、物理严谨性和推理效率之间的瓶颈，我们提出广义神经算子。通过在神经算子中形式化适定性的经典条件，我们的框架展示了显式依赖PDE参数和边界条件的理论优势。为在不影响计算速度的情况下实现这种综合，我们引入三个新颖的架构组件：用于高效参数泛化的参数门控核混合、将任意边界约束投影到统一潜在狄利克雷表示的广义边界传递算子以及确保稳定性的专门训练目标。广泛实验表明，我们基于理论的方法在跨异构物理状态实现卓越泛化的同时，保持与传统数值基线相当的严格推理效率。
英文摘要
Developing foundational neural simulators for Partial Differential Equations (PDEs) requires robust generalization across diverse physical parameters and boundary conditions. However, current deep learning approaches largely face a structural trade-off between condition-agnostic deployment and physical fidelity. Purely data-driven operators infer the underlying physics implicitly and thus lack the explicit constraints needed to ensure physically valid solutions across varying domains, rendering the learning problem ill-posed. On the other hand, Physics-Informed Neural Networks (PINNs) enforce rigorous physical constraints but necessitate costly, instance-specific optimization. Furthermore, the massive scale of emerging foundational operators has severely degraded their inference speeds, making them computationally uncompetitive with traditional numerical solvers. To address this bottleneck between condition-agnostic deployment, physical rigor, and inference efficiency, we propose a \textit{Generalized Neural Operator}. By formalizing the classical conditions for well-posedness within neural operators, our framework demonstrates the theoretical benefits of explicitly conditioning on PDE parameters and boundary conditions. To implement this synthesis without compromising computational speed, we introduce three novel architectural components: a parameter-gated mixture of kernels for efficient parameter generalization, a generalized boundary transfer operator that projects arbitrary boundary constraints into a unified latent Dirichlet representation, and a specialized training objective to ensure stability. Extensive experiments demonstrate that our theoretically grounded approach achieves superior generalization across heterogeneous physical regimes while maintaining strict inference efficiency comparable to conventional numerical baselines.
