# PE-MHL: Physics-Encoded Modular Hybrid Layers for Scalable Learning of Complex Systems

arXiv: https://arxiv.org/abs/2606.04290 · HTML: https://mlsanity.com/paper/2606.04290/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 16 references that are in the map.
- First posted on arXiv 2 Jun 2026.

## Builds on (most cited first, 7 of 7)

- [Overcoming catastrophic forgetting in neural networks](https://mlsanity.com/paper/1612.00796/) (2016)
- [Physics-Guided Machine Learning for Scientific Discovery: An Application in Simulating Lake Temperature Profiles](https://mlsanity.com/paper/2001.11086/) (2020)
- [Physics-Guided Neural Networks for Inversion-based Feedforward Control applied to Linear Motors](https://mlsanity.com/paper/2103.06092/) (2021)
- [Response Estimation and System Identification of Dynamical Systems via Physics-Informed Neural Networks](https://mlsanity.com/paper/2410.01340/) (2024)
- [Robustness of Physics-Informed Neural Networks to Noise in Sensor Data](https://mlsanity.com/paper/2211.12042/) (2022)
- [Control-Oriented System Identification: Classical, Learning, and Physics-Informed Approaches](https://mlsanity.com/paper/2512.06315/) (2025)
- [On feedforward control using physics-guided neural networks: Training cost regularization and optimized initialization](https://mlsanity.com/paper/2201.12088/) (2022)

## Abstract (arXiv)

Hybrid models that combine physics-based and data-driven components have shown strong potential for achieving accuracy and interpretability in control applications. While recent methods have made progress in incorporating physical consistency, challenges remain in scalability, robustness to noise, and control of model complexity. This paper proposes a Physics-Encoded Modular Hybrid Layer (PE-MHL) framework, in which a baseline physics-based model is incrementally refined through the addition of new sub-models, where each new component adds complexity while preserving what previous components have already learned. We establish a theoretical guarantee for this construction: with a least-squares initialization of each new sub-model, the training error is monotonically non-increasing in the number of sub-models and provably converges. Empirical evaluations on a nonlinear NARX benchmark and the Quanser Aero 2 platform demonstrate that PE-MHL outperforms equivalently sized monolithic networks in both accuracy and generalization, while also providing more stable training dynamics and better preservation of underlying data structures.