Physics-Integrated Operator Learning Improves PDE System Prediction

Neural operators are efficient tools for simulating spatiotemporal Partial Differential Equation (PDE) systems, but purely data-driven versions can accumulate errors over long prediction horizons and may not fully leverage known governing equations. Existing methods for physics integration often involve residual-based training or PDE-specific architectural constraints, which can complicate optimization or limit architectural flexibility. Researchers have developed a new representation-level approach to integrate physics into neural operators. This method utilizes a feed-forward Gaussian splatting (FFGS) representation, which acts as a continuous interface connecting discretized solution fields with the governing operators. The FFGS representation reconstructs the state as a continuous Gaussian field, providing closed-form spatial derivatives. This allows known physical PDE operators to be directly incorporated into the learned evolution map without needing a separate physics-residual loss. Evaluated across 2D and 3D PDE systems, including advection, diffusion, and reaction dynamics, the framework reduced relative L2 error by 1.5x to 2.2x compared to data-driven baselines during long-horizon autoregressive rollouts, while consistently enhancing spectral fidelity. The approach also proved effective even when governing equations were only partially known, demonstrating robustness.