Structure-Preserving Digital Twins via Conditional Neural Whitney Forms

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.

Natively Interpretable Machine Learning and Artificial Intelligence: Preliminary Results and Future Directions

Machine learning models have become more and more complex in order to better approximate complex functions. Although fruitful in many domains, the added complexity has come at the cost of model interpretability. The once popular k-nearest neighbors (kNN) approach, which finds and uses the most similar data for reasoning, has received much less attention in recent decades due to numerous problems when compared to other techniques. We show that many of these historical problems with kNN can be overcome, and our contribution has applications not only in machine learning but also in online learning, data synthesis, anomaly detection, model compression, and reinforcement learning, without sacrificing interpretability. We introduce a synthesis between kNN and information theory that we hope will provide a clear path towards models that are innately interpretable and auditable. Through this work we hope to gather interest in combining kNN with information theory as a promising path to fully auditable machine learning and artificial intelligence.