Physics-Integrated Operator Learning via Gaussian Splatting Representations

2026-08-25

AI Summary

该研究通过引入前馈高斯涂抹表示法，将物理方程直接整合到学习的演化映射中，解决了数据驱动方法在长时间预测中的误差累积问题。

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.

Problem

• neural operators
• spatiotemporal PDE systems
• autoregressive prediction
• governing-equation structure
• Gaussian splatting

Innovation

• Physics-Integrated Operator Learning
• Gaussian Splatting Representations
• Continuous Interface
• Long-Horizon Autoregressive Prediction
• Physical PDE Operators