# MINN: Learning the dynamics of differential-algebraic equations and application to battery modeling

arXiv: https://arxiv.org/abs/2304.14422 · HTML: https://mlsanity.com/paper/2304.14422/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 69 references that are in the map.
- First posted on arXiv 27 Apr 2023.

## Builds on (most cited first, 12 of 12)

- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Revealing hidden dynamics from time-series data by ODENet](https://mlsanity.com/paper/2005.04849/) (2020)
- [Predicting battery end of life from solar off-grid system field data using machine learning](https://mlsanity.com/paper/2107.13856/) (2021)
- [High-performance symbolic-numerics via multiple dispatch](https://mlsanity.com/paper/2105.03949/) (2021)
- [Lithium-ion battery thermal-electrochemical model-based state estimation using orthogonal collocation and a modified extended Kalman filter](https://mlsanity.com/paper/1506.08689/) (2015)

## Built on by (most cited first, 10 of 10)

- [Learning Neural Differential Algebraic Equations via Operator Splitting](https://mlsanity.com/paper/2403.12938/) (2024)
- [Physics-informed machine learning of redox flow battery based on a two-dimensional unit cell model](https://mlsanity.com/paper/2306.01010/) (2023)
- [A Simultaneous Approach for Training Neural Differential-Algebraic Systems of Equations](https://mlsanity.com/paper/2504.04665/) (2025)
- [Hard-Constrained Neural Networks with Physics-Embedded Architecture for Residual Dynamics Learning and Invariant Enforcement in Cyber-Physical Systems](https://mlsanity.com/paper/2511.23307/) (2025)
- [Neural Port-Hamiltonian Differential Algebraic Equations for Compositional Learning of Electrical Networks](https://mlsanity.com/paper/2412.11215/) (2024)
- [Diagnostic-free onboard battery health assessment](https://mlsanity.com/paper/2503.07383/) (2025)
- [Fast and Generalizable parameter-embedded Neural Operators for Lithium-Ion Battery Simulation](https://mlsanity.com/paper/2508.08087/) (2025)
- [Learning to Test: Latent Representation Learning for Dynamic Instability Detection](https://mlsanity.com/paper/2604.10967/) (2026)
- [Comparative Study of Neural Surrogate Architectures for Autoregressive Prediction of Internal Battery States](https://mlsanity.com/paper/2606.20053/) (2026)
- [Bridging battery design and health assessment through virtual sensing and physics-informed learning](https://mlsanity.com/paper/2607.16864/) (2026)

## Abstract (arXiv)

The concept of integrating physics-based and data-driven approaches has become popular for modeling sustainable energy systems. However, the existing literature mainly focuses on the data-driven surrogates generated to replace physics-based models. These models often trade accuracy for speed but lack the generalizability, adaptability, and interpretability inherent in physics-based models, which are often indispensable in modeling real-world dynamic systems for optimization and control purposes. We propose a novel machine learning architecture, termed model-integrated neural networks (MINN), that can learn the physics-based dynamics of general autonomous or non-autonomous systems consisting of partial differential-algebraic equations (PDAEs). The obtained architecture systematically solves an unsettled research problem in control-oriented modeling, i.e., how to obtain optimally simplified models that are physically insightful, numerically accurate, and computationally tractable simultaneously. We apply the proposed neural network architecture to model the electrochemical dynamics of lithium-ion batteries and show that MINN is extremely data-efficient to train while being sufficiently generalizable to previously unseen input data, owing to its underlying physical invariants. The MINN battery model has an accuracy comparable to the first principle-based model in predicting both the system outputs and any locally distributed electrochemical behaviors but achieves two orders of magnitude reduction in the solution time.