[Paper Review] Operator Learning with Neural Fields: Tackling PDEs on General Geometries

Louis Serrano, Lise Le Boudec+5 more|arXiv (Cornell University)|Jun 12, 2023

Model Reduction and Neural NetworksPhysics and Astronomy51 references15 citations

TL;DR

CORAL introduces coordinate-based neural field operators that map between functions on general geometries, enabling PDE solving on irregular meshes and showing strong versatility across IVP, dynamics, and geometry-aware inference tasks.

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.

Motivation & Objective

• Motivate learning mappings between infinite-dimensional function spaces to solve PDEs on general geometries.
• Develop a flexible operator-learning framework that avoids mesh constraints and supports irregular sampling.
• Demonstrate CORAL across multiple physics-informed tasks (initial value problems, dynamics forecasting, geometry-aware inference).
• Show that using coordinate-based representations yields robustness to unseen grids and geometries.

Proposed method

• Use two modulated implicit neural representations (INRs) to encode input and output functions with shared base parameters and per-function modulations.
• Encode inputs into a low-dimensional latent code via a spatial encoder and auto-decoding through an INR with modulation.
• Process latent codes with a potentially flexible processor (e.g., simple MLP or Neural ODE solver) to map input latent to output latent.
• Decode the output latent code into a spatial function using a modulated INR to obtain predictions at any query location.
• Train with a two-step procedure: first fit INRs to reconstruct inputs/outputs, then train the latent-space processor.
• Adopt SIREN-based INRs with shift modulations to balance expressivity and efficiency.

Experimental results

Research questions

• RQ1 Can CORAL learn mappings between functions sampled on irregular or unseen grids without mesh constraints?
• RQ2 How does CORAL perform across diverse tasks: initial value problems, dynamics forecasting, and geometry-aware inference?
• RQ3 Is CORAL robust to changes in spatial sampling and geometry compared to existing operator or mesh-based models?
• RQ4 Does the two-step training with latent codes enable fast and effective learning of complex operators?
• RQ5 How does CORAL's performance compare to state-of-the-art baselines on convex and non-convex domains and on planar versus spherical geometries?

Key findings

• CORAL achieves competitive or state-of-the-art performance across IVP, dynamics forecasting, and geometry-aware inference tasks.
• The model generalizes well to new grids not seen during training, including highly irregular sampling and non-convex domains.
• CORAL maintains strong performance under spatial sparsity and variations in mesh granularity, often outperforming alternative mesh-based or fixed-grid operators.
• Inference is fast due to compact latent representations enabling efficient decoding of spatial functions.
• Compared to several baselines, CORAL shows robustness to grid changes and irregular sampling where some methods overfit to the training grid.