Title: Flow Learners for PDEs: Toward a Physics-to-Physics Paradigm for Scientific Computing

Abstract:

Partial differential equations (PDEs) underpin the vast majority of physical phenomena studied in science and engineering; however, solving these equations at a large scale remains computationally prohibitive. While generative AI has revolutionized fields such as language processing, computer vision, and protein science, the domain of learned PDE solvers has not experienced a similar transformation. Current methodologies each address only fragments of the challenge. Physics-informed neural networks incorporate residual structures but often struggle with optimization in contexts involving stiffness, multiscale dynamics, or extensive domains. Neural operators offer amortization across different instances but typically retain a snapshot-prediction perspective, leading to performance degradation during extended rollouts. Meanwhile, diffusion-based solvers are capable of modeling uncertainty yet frequently rely on solver templates that remain anchored in state regression.

We posit that the fundamental limitation lies in the abstraction employed to train these learned solvers. Many existing models are tasked with predicting static states, whereas numerous scientific applications demand the modeling of how uncertainty propagates through constrained dynamics. The essential object of study is therefore transport over physically admissible futures. This insight gives rise to "flow learners": models that parameterize transport vector fields and produce trajectories via integration, thereby mirroring the continuous dynamics that characterize PDE evolution. This alignment between physics and physics facilitates continuous-time prediction, enables native uncertainty quantification, and opens new avenues for designing physics-aware solvers. In this work, we elucidate why transport-based learning serves as a more robust organizing principle for learned PDE solving and delineate the research agenda that emerges from this paradigm shift.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC