Geometry aware inference of steady state PDEs using Equivariant Neural Fields representations

2025-04-24

AI Summary

This work addresses the problem of solving steady-state partial differential equations (PDEs) on geometrically variable domains. We propose the *enf2enf* encoder-decoder framework: an equivariant point-cloud encoder maps input geometries into equivariant embeddings, which—after fusion with global parameters—are directly decoded into continuous, coordinate-based neural fields representing physical solutions. Our approach introduces the first geometry-aware equivariant neural field representation, implicitly capturing strong geometry–physics coupling while preserving locality and translational invariance as inductive biases. It achieves both local physical fidelity and robustness to global shape deformation. The method enables zero-shot super-resolution and real-time inference, delivering full-scale, high-fidelity predictions from low-resolution training data alone. Evaluated on high-fidelity aerodynamic, hyperelastic material, and multi-airfoil benchmarks, it establishes new state-of-the-art performance.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Computer Vision: Low Level & Physics-based Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search

Abstract

Recent advances in Neural Fields have enabled powerful, discretization-invariant methods for learning neural operators that approximate solutions of Partial Differential Equations (PDEs) on general geometries. Building on these developments, we introduce enf2enf, an encoder--decoder methodology for predicting steady-state Partial Differential Equations with non-parameterized geometric variability, based on recently proposed Equivariant Neural Field architectures. In enf2enf, input geometries are encoded into latent point cloud embeddings that inherently preserve geometric grounding and capture local phenomena. The resulting representations are then combined with global parameters and directly decoded into continuous output fields, thus efficiently modeling the coupling between geometry and physics. By leveraging the inductive biases of locality and translation invariance, our approach is able to capture fine-scale physical features as well as complex shape variations, thereby enhancing generalization and physical compliance. Extensive experiments on a high-fidelity aerodynamic dataset, a hyper-elastic material benchmark, and multi-element airfoil geometries, demonstrate that the proposed model achieves superior or competitive performance compared to state-of-the-art graph based, operator learning, and neural field methods. Notably, our method supports real time inference and zero-shot super-resolution, enabling efficient training on low-resolution meshes while maintaining high accuracy on full-scale discretizations.

Problem

• Predict steady-state PDEs with non-parameterized geometric variability
• Encode input geometries into latent point cloud embeddings
• Achieve real-time inference and zero-shot super-resolution

Innovation

• Equivariant Neural Fields for PDE solutions
• Encoder-decoder with latent point clouds
• Real-time inference and super-resolution