Continuous PDE Dynamics Forecasting with Implicit Neural Representations

Yuan Yin, Matthieu Kirchmeyer, Jean-Yves Franceschi, Alain Rakotomamonjy, patrick gallinari

Keywords: spatiotemporal forecasting, Partial Differential Equations, PDEs, Implicit Neural Representations, INRs, continuous models, generalization, dynamical systems, physics

TL;DR: We propose a continuous-time, continuous-space data-driven PDE forecasting model with extensive spatiotemporal extrapolation capabilities including generalization to unseen sparse meshes and resolutions.

Abstract: Effective data-driven PDE forecasting methods often rely on fixed spatial and / or temporal discretizations. This raises limitations in real-world applications like weather prediction where flexible extrapolation at arbitrary spatiotemporal locations is required. We address this problem by introducing a new data-driven approach, DINo, that models a PDE's flow with continuous-time dynamics of spatially continuous functions. This is achieved by embedding spatial observations independently of their discretization via Implicit Neural Representations in a small latent space temporally driven by a learned ODE. This separate and flexible treatment of time and space makes DINo the first data-driven model to combine the following advantages. It extrapolates at arbitrary spatial and temporal locations; it can learn from sparse irregular grids or manifolds; at test time, it generalizes to new grids or resolutions. DINo outperforms alternative neural PDE forecasters in a variety of challenging generalization scenarios on representative PDE systems.

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