Learning Space-Time Continuous Neural PDEs from Partially Observed States

TL;DR

This paper presents a grid-independent neural PDE model that learns from noisy, partial observations on irregular spatiotemporal grids, combining probabilistic inference and novel encoder design for improved data efficiency and stability.

Contribution

It introduces a space-time continuous latent neural PDE framework with a new encoder, combining collocation and method of lines, and employs variational inference and multiple shooting for robust training.

Findings

Achieves state-of-the-art results on synthetic and real datasets.

Handles partially observed data effectively.

Demonstrates robustness and grid independence in modeling complex dynamics.

Abstract

We introduce a novel grid-independent model for learning partial differential equations (PDEs) from noisy and partial observations on irregular spatiotemporal grids. We propose a space-time continuous latent neural PDE model with an efficient probabilistic framework and a novel encoder design for improved data efficiency and grid independence. The latent state dynamics are governed by a PDE model that combines the collocation method and the method of lines. We employ amortized variational inference for approximate posterior estimation and utilize a multiple shooting technique for enhanced training speed and stability. Our model demonstrates state-of-the-art performance on complex synthetic and real-world datasets, overcoming limitations of previous approaches and effectively handling partially-observed data. The proposed model outperforms recent methods, showing its potential to advance data-driven PDE modeling and enabling robust, grid-independent modeling of complex partially-observed dynamic processes.