Precise Gradient Discontinuities in Neural Fields for Subspace Physics

2025-05-26

AI Summary

Existing neural fields struggle to model spatial derivative discontinuities—such as creases or abrupt stiffness changes—in physical systems without prior knowledge or adaptive mesh remeshing. Method: We propose a coordinate lifting mechanism based on a smoothed truncation distance function, which explicitly encodes gradient jumps while preserving overall function continuity. This enables end-to-end learning and geometry-aware embedding of discontinuity locations. Our framework integrates neural implicit fields with hybrid soft-rigid parameterization, supporting unified modeling of both value and gradient discontinuities, as well as dynamic interface evolution. Contribution/Results: Experiments demonstrate generalization across diverse shape families, enabling interactive crease editing, real-time morphological interpolation, and soft–hard coupled simulation—all without mesh remeshing. The approach significantly enhances the representational capacity and simulation flexibility of reduced-order models.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs

Abstract

Discontinuities in spatial derivatives appear in a wide range of physical systems, from creased thin sheets to materials with sharp stiffness transitions. Accurately modeling these features is essential for simulation but remains challenging for traditional mesh-based methods, which require discontinuity-aligned remeshing -- entangling geometry with simulation and hindering generalization across shape families. Neural fields offer an appealing alternative by encoding basis functions as smooth, continuous functions over space, enabling simulation across varying shapes. However, their smoothness makes them poorly suited for representing gradient discontinuities. Prior work addresses discontinuities in function values, but capturing sharp changes in spatial derivatives while maintaining function continuity has received little attention. We introduce a neural field construction that captures gradient discontinuities without baking their location into the network weights. By augmenting input coordinates with a smoothly clamped distance function in a lifting framework, we enable encoding of gradient jumps at evolving interfaces. This design supports discretization-agnostic simulation of parametrized shape families with heterogeneous materials and evolving creases, enabling new reduced-order capabilities such as shape morphing, interactive crease editing, and simulation of soft-rigid hybrid structures. We further demonstrate that our method can be combined with previous lifting techniques to jointly capture both gradient and value discontinuities, supporting simultaneous cuts and creases within a unified model.

Problem

• Modeling gradient discontinuities in neural fields for physics
• Enabling simulation of heterogeneous materials with sharp transitions
• Supporting shape morphing and interactive crease editing

Innovation

• Neural field with gradient discontinuity encoding
• Smoothly clamped distance function augmentation
• Discretization-agnostic simulation for shape families