Structure-Preserving Digital Twins via Conditional Neural Whitney Forms

TL;DR

This paper introduces a structure-preserving neural framework for real-time digital twins that guarantees conservation laws, supports complex geometries, and achieves high accuracy and speedup in diverse physical simulations.

Contribution

It develops a novel conditional neural Whitney form approach within finite element exterior calculus that ensures numerical well-posedness and conservation, enabling real-time, calibrated digital twins.

Findings

Achieves real-time inference in 0.1 seconds.

Maintains conservation laws regardless of data sparsity.

Demonstrates high accuracy on complex physical problems.

Abstract

We present a framework for constructing real-time digital twins based on structure-preserving reduced finite element models conditioned on a latent variable Z. The approach uses conditional attention mechanisms to learn both a reduced finite element basis and a nonlinear conservation law within the framework of finite element exterior calculus (FEEC). This guarantees numerical well-posedness and exact preservation of conserved quantities, regardless of data sparsity or optimization error. The conditioning mechanism supports real-time calibration to parametric variables, allowing the construction of digital twins which support closed loop inference and calibration to sensor data. The framework interfaces with conventional finite element machinery in a non-invasive manner, allowing treatment of complex geometries and integration of learned models with conventional finite element techniques. Benchmarks include advection diffusion, shock hydrodynamics, electrostatics, and a complex battery thermal runaway problem. The method achieves accurate predictions on complex geometries with sparse data (25 LES simulations), including capturing the transition to turbulence and achieving real-time inference ~0.1s with a speedup of 3.1x10^8 relative to LES. An open-source implementation is available on GitHub.