Operator Learning with Neural Fields: Tackling PDEs on General Geometries

Abstract

Machine learning approaches for solving partial differential equations require learning mappings between function spaces. While convolutional or graph neural networks are constrained to discretized functions, neural operators present a promising milestone toward mapping functions directly. Despite impressive results they still face challenges with respect to the domain geometry and typically rely on some form of discretization. In order to alleviate such limitations, we present CORAL, a new method that leverages coordinate-based networks for solving PDEs on general geometries. CORAL is designed to remove constraints on the input mesh, making it applicable to any spatial sampling and geometry. Its ability extends to diverse problem domains, including PDE solving, spatio-temporal forecasting, and inverse problems like geometric design. CORAL demonstrates robust performance across multiple resolutions and performs well in both convex and non-convex domains, surpassing or performing on par with state-of-the-art models.

References in corpus (7)

• Instant Neural Graphics Primitives with a Multiresolution Hash Encoding
• Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains
• Meta-SGD: Learning to Learn Quickly for Few-Shot Learning
• Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge
• MetaSDF: Meta-learning Signed Distance Functions
• GNOT: A General Neural Operator Transformer for Operator Learning
• Physical Design using Differentiable Learned Simulators

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