arXiv：cs.LG· Jihao Zhang, Junyi Guo, Jian-Xun Wang·· 4 小时前
基于高斯泼溅表示的物理融合算子学习
Physics-Integrated Operator Learning via Gaussian Splatting Representations
AI 导读
研究提出前馈高斯泼溅（FFGS）表示，作为离散解场与支配算子之间的连续接口，将物理PDE算子直接融入学习到的演化映射，无需引入物理残差损失。在二维和三维PDE系统（平流、扩散、非线性自平流和反应动力学）的长期自回归预测中，该方法较最强纯数据驱动基线将相对 $\ell_2$ 误差降低 1.5 倍至 2.2 倍，并持续提升谱保真度；在支配方程部分已知时仍保持有效。
正文
Abstract: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.
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2608.24049 [cs.LG] |
| (or arXiv:2608.24049v2 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2608.24049 arXiv-issued DOI via DataCite |
Submission history
From: Jian-Xun Wang [view email]
[v1]
Tue, 25 Aug 2026 04:18:08 UTC (7,085 KB)
[v2]
Wed, 14 Oct 2026 12:40:23 UTC (7,056 KB)
来源：arXiv：cs.LG · arxiv.org