arXiv:2512.21311v2 Announce Type: replace Abstract: Solving partial differential equations (PDEs) on shapes underpins many shape analysis and engineering tasks; yet, prevailing PDE solvers operate on polygonal/triangle meshes while modern 3D assets increasingly live as neural representations. This mismatch leaves no suitable method to solve surface PDEs directly within the neural domain, forcing explicit mesh extraction or per-instance residual training, preventing end-to-end workflows. We present a novel, meshfree formulation that learns a local update operator conditioned on neural (local) s
This development arises as neural representations become the standard for 3D assets, creating a growing mismatch with traditional PDE solvers.
This innovation significantly advances the capability to perform highly complex engineering simulations directly on modern AI-driven 3D models, accelerating design and analysis in numerous fields.
PDEs can now be solved directly on neural shape representations without intermediate mesh extraction or per-instance training, enabling end-to-end workflows in the neural domain.
- · AI/ML researchers in 3D modeling
- · Engineering design firms
- · Digital twin developers
- · Manufacturing and aerospace sectors
- · Developers of traditional mesh-based PDE solvers
- · Workflows reliant on explicit mesh extraction
Engineers and designers gain more efficient and accurate simulation tools for complex geometries represented by neural networks.
This could lead to a significant acceleration in R&D cycles for products with intricate designs, from aerospace components to medical devices.
The integration of simulation with neural representations may catalyze more sophisticated autonomous design and optimization agents.
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Read at arXiv cs.LG