Precise Gradient Discontinuities in Neural Fields for Subspace Physics

TL;DR

This work tackles gradient discontinuities in neural field representations for physics simulation by introducing a lifting framework that augments input coordinates with a smoothly clamped distance function $H$, producing a discretization-agnostic reduced-space basis capable of handling gradient jumps at evolving interfaces. The method enables simulations across parametric shape and material spaces, supports interactive crease editing, and can be combined with winding-number lifting to model both function- and gradient-discontinuities within a unified model. It demonstrates competitive accuracy with FEM for gradient-discontinuous interfaces while maintaining generalization across shapes and materials and enabling differentiable shape optimization in reduced space. Overall, the approach integrates gradient discontinuity modeling into neural fields, unlocking efficient, generalizable subspace simulations for heterogeneous materials and crease-daden geometries with potential for broader applications in design and real-time deformation analysis.

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.

Figures (15)

• Figure 1: Our method represents functions with discontinuous gradients by lifting the input domain into a higher-dimensional space. Starting from an input domain with internal interfaces, we construct a smoothly clamped distance field to augment the spatial coordinates. This defines a lifted domain where a neural network is trained to produce smooth basis functions. When restricted back to the original domain, the resulting basis captures sharp gradient transitions at the interface.
• Figure 2: We visualize the smoothly clamped distance function and its gradient. The function flattens beyond a threshold distance $s$, while preserving gradient discontinuities at the interface (where the distance is zero).
• Figure 3: We test our model on a complex scene of a hand with soft flesh and a stiff skeleton. This example demonstrates both reduced memory usage and speedup enabled by our spatial hash. The clamped distance function localizes queries by ignoring point pairs farther than $s$, making it well-suited for spatial hashing. The hash structure supports 4.1$\times$ more queries due to improved memory efficiency, and achieves a 3.6$\times$ speedup when tested with 90k query points (same as without hashing).
• Figure 4: We compared our method against basis functions produced by other neural field architectures on a heterogeneous 2D "U"-shaped domain. For visualization, the 2D shape is lifted along the Y-axis to represent the scalar basis function, with additional color coding to indicate its value. The SIREN MLP Sitzmann:2020:Implicit, used in prior works chang2024neuralrepresentationshapedependentlaplacianModi:2024:Simplicits, fails to capture sharp variations at the material interface. A ReLU-based neural field, which permits $C^0$ continuity, does not converge to the correct solution. In contrast, our method successfully captures the sharp gradient transitions across the interface.
• Figure 5: Simulation of a parameterized shape family morphing from a fox to a bear, performing nodding and shaking motions. Each shape includes a skull that is $100 \times$ stiffer than the surrounding soft tissue, shown in the X-ray view. Our neural basis adapts across the family: soft regions (ears, nose) undergo large deformations, while the stiff skull remains rigid.