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

TL;DR

The paper introduces model-integrated neural networks (MINN), a physics-constrained learning framework that embeds PDAE dynamics into a neural recurrent unit by approximating algebraic variables with a network $G_{NN}$ and solving $egin{cases}\dot{h}_d=f(t,h_d,h_z,u)\ h_z^*=G_{NN}(t,h_d,u; heta)\ 0=g(t,h_d,h_z,u)\\, ext{(plus outputs)} \ar{g}=g(t,h_d,h_z,u)\ ext{(evaluated at each step)} \end{cases}$ to yield a data-efficient surrogate. Applied to lithium-ion batteries via the P2D PDAE framework, MINN uses orthogonal collocation for spatial discretization and learns with a physics-constrained loss $\,\mathcal{L}_{MINN}=\mathcal{L}_{y}+\lambda\mathcal{L}_{g}$, enabling accurate predictions of terminal voltage, SOC, plating potential, and local electrochemical fields while achieving substantial speedups over full PDAE solutions. Compared with DNN, PINN, NODE, and DD-ROM baselines, MINN delivers superior generalization to unseen control inputs, maintains physically meaningful hidden states, and reduces computational cost by about two orders of magnitude, with an adjustable order (e.g., 82 vs 130 states) without sacrificing accuracy. The approach holds promise for adaptive, aging-aware battery management and general non-autonomous PDAEs, offering real-time capability for optimization, control, and safety prognostics in energy systems and beyond.

Abstract

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.

Figures (7)

• Figure 1: Existing physics-based integration strategies for the blending of neural networks and physics-based models in order to retain their individual merits. (a) A data-driven surrogate model using supervised learning requires relevant and representative training data generated by snapshots of the physics-based model solutions. (b) A surrogate model regularised physical constraints within the PINN framework, of which the PINN loss, $\mathcal{L}_{\text{PINN}}$, is composed of the loss due to model-data inconsistency, $\mathcal{L}_{\text{surrogate}}$, and the loss owing to physical constraints, $\mathcal{L}_{\text{physical}}$. (c) The PINN workflow for inverse problems used to estimate physical parameters as part of parametric PDEs Raissi_2017_2.
• Figure 2: The proposed MINN architecture for dynamic systems. (a) An iterative update of the hidden states $h^k_d$, output $y^k$ and conserved quantities $\bar{g}^k$, is controlled by input $u^k=u(t^k)$ at time $t^k$ through physics-based hidden units. This update is handled by the time integration via the numerical solver. (b) The design of a physics-based recurrent unit contains physics-based equations, a deep learning-enabled approximation \ref{['Eqn. DAE08']} and an output function $Y$.
• Figure 3: The realization of the "physics-based equations" module in the physics-based recurrent unit of Fig. \ref{['Fig:MINN']}. For battery systems, the control input $u$ is the applied current, i.e. $u^k = I(t^k)$, and the differential state and algebraic variables are represented by $h_d=[C_s,\,C_e]^T$ and $h_z=j$, respectively. The $g$-component evaluates the conservation laws at each time step $k$ with the approximated algebraic variable $h_z^{*}$, while the $f$-component evaluates the time derivative of the differential states $\dot{h}_d$. The two components in the circuitry feature P2D equations, e.g., the open circuit potential (OCP) is a fitted function that takes in the solid concentration at the active material surface and outputs the equilibrium potential $\phi_{eq}$.
• Figure 4: P2D representation of a LIB cell with superimposed continua spanning over two phases and three domains. The nomenclature can be found in the Supplementary Information.
• Figure 5: Comparison of different data-driven and hybrid model predictions under 1C charge with P2D predictions as ground truth. All four models are trained using data sampled from the first 300 seconds of the 1C charge data starting from 30% SOC. (a)--(e) Spatiotemporal plots of electrolyte concentration. (f) The mean absolute percentage error of electrolyte concentration averaged over the thickness. (g) The root mean square error of the plating potential averaged over the thickness. (h)-(l) Spatiotemporal plots of anode potential $\phi_s - \phi_e$.