发表机构
Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出基于前馈高斯溅射（FFGS）表示的物理集成算子学习框架，在多类PDE系统长时序预测中降低相对ℓ₂误差1.5-2.2倍，提升光谱保真度，对部分已知控制方程具鲁棒性。
AI 中文摘要
神经算子可为时空偏微分方程（PDE）系统提供高效的替代模型，但纯数据驱动的公式在长时序自回归预测中往往会积累大量误差，且无法利用已有的控制方程结构。现有方法主要通过基于残差的训练目标或PDE特定的架构约束来融入物理信息，这可能会带来优化困难或限制架构的通用性。在本研究中，我们提出一种表示层面的物理集成方法，其中前馈高斯溅射（FFGS）表示作为离散解场与控制算子之间的连续接口。FFGS表示将状态重构为具有闭式空间导数的连续高斯场，允许在学习的演化映射中直接集成可用的物理PDE算子，而无需引入物理残差损失。我们在二维和三维PDE系统（包括平流、扩散、非线性自平流和反应动力学）上对该框架进行了评估。在长时序自回归滚动预测中，与基准套件中最强的纯数据驱动基线相比，所提框架将相对ℓ₂误差降低了1.5倍至2.2倍，同时持续提升了光谱保真度。当控制方程部分已知时，该框架仍能保持有效性，展现出对不完整物理信息的鲁棒性。这些结果表明，连续场表示可为将已知物理结构融入通用神经算子替代模型提供实用接口。
英文摘要
Neural operators provide efficient surrogates for spatiotemporal PDE systems, but purely data-driven formulations often accumulate substantial errors during long-horizon autoregressive prediction and may fail to exploit available governing-equation structure. Existing approaches incorporate physics primarily through residual-based training objectives or PDE-specific architectural constraints, which can introduce optimization difficulties or limit architectural generality. In this work, we introduce a representation-level approach to physics integration in which a feed-forward Gaussian splatting (FFGS) representation serves as a continuous interface between discretized solution fields and governing operators. The FFGS representation reconstructs the state as a continuous Gaussian field with closed-form spatial derivatives, allowing available physical PDE operators to be integrated directly within the learned evolution map without introducing a physics-residual loss. We evaluate the framework across two- and three-dimensional PDE systems, including advection, diffusion, nonlinear self-advection, and reaction dynamics. Over long-horizon autoregressive rollouts, the proposed framework reduces relative $\ell_2$ error by $1.5\times$--$2.2\times$ compared with the strongest purely data-driven baseline across the benchmark suite, while consistently improving spectral fidelity. The framework also remains effective when the governing equations are partially known, demonstrating robustness to incomplete physics. These results demonstrate that continuous field representations can provide a practical interface for incorporating known physical structure into generic neural-operator surrogates.
