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This study introduces a novel feed-forward Gaussian splatting (FFGS) representation that integrates physics directly into neural operators for spatiotemporal PDE systems, addressing the limitations of traditional data-driven methods that struggle with long-horizon predictions. By reconstructing the state as a continuous Gaussian field with closed-form spatial derivatives, the FFGS representation allows for the direct incorporation of governing PDE operators without relying on physics-residual losses. The framework shows a significant reduction in relative $\ell_2$ error by 1.5 to 2.2 times compared to leading data-driven approaches, while enhancing spectral fidelity and demonstrating robustness even with partially known governing equations.
Integrating physics directly into neural operators can cut prediction errors by up to 2.2 times, transforming how we model complex dynamical systems.
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.
