Domain-decoupled Physics-informed Neural Networks with Closed-form Gradients for Fast Model Learning of Dynamical Systems

TL;DR

The paper tackles slow training and instability in physics-informed neural networks for control (PINCs) by introducing the domain-decoupled PINN (DD-PINN), which decouples time from the neural network to yield closed-form gradients for the physics loss. The key idea is an Ansatz $\hat{\bm{x}}_t = \bm{g}(\bm{a},t) + \bm{x}_0$ with $\bm{a}=\bm{f}_\mathrm{NN}(\bm{x}_0,\bm{u}_0,\boldsymbol{\theta})$ so that $\partial \hat{\bm{x}}_t/\partial t = \dot{\bm{g}}(\bm{a},t)$ and $\bm{g}(\bm{a},0)=\bm{0}$, which eliminates the initial-condition loss and enables analytical gradient computation. The method also supports higher-order excitation inputs to overcome zero-order-hold limitations. Across a nonlinear mass–spring–damper, a five-mass chain, and a two-link manipulator, DD-PINN achieves substantially faster training and more stable, accurate self-loop predictions than PINCs, enabling fast surrogate learning suitable for real-time control and MPC. This approach preserves PINC compatibility while offering broad applicability to large-scale dynamical systems with improved data efficiency and robustness.

Abstract

Physics-informed neural networks (PINNs) are trained using physical equations and can also incorporate unmodeled effects by learning from data. PINNs for control (PINCs) of dynamical systems are gaining interest due to their prediction speed compared to classical numerical integration methods for nonlinear state-space models, making them suitable for real-time control applications. We introduce the domain-decoupled physics-informed neural network (DD-PINN) to address current limitations of PINC in handling large and complex nonlinear dynamical systems. The time domain is decoupled from the feed-forward neural network to construct an Ansatz function, allowing for calculation of gradients in closed form. This approach significantly reduces training times, especially for large dynamical systems, compared to PINC, which relies on graph-based automatic differentiation. Additionally, the DD-PINN inherently fulfills the initial condition and supports higher-order excitation inputs, simplifying the training process and enabling improved prediction accuracy. Validation on three systems - a nonlinear mass-spring-damper, a five-mass-chain, and a two-link robot - demonstrates that the DD-PINN achieves significantly shorter training times. In cases where the PINC's prediction diverges, the DD-PINN's prediction remains stable and accurate due to higher physics loss reduction or use of a higher-order excitation input. The DD-PINN allows for fast and accurate learning of large dynamical systems previously out of reach for the PINC.

Figures (9)

• Figure 1: Configurations for (a) PINC training and (b) PINC self-loop prediction process as well as (c) DD-PINN training and (d) DD-PINN self-loop prediction. Time domain is decoupled from the FNN for the DD-PINN, enabling calculation of gradients in closed form.
• Figure 2: Three dynamical systems for evaluation: (a) nonlinear mass-spring-damper system, (b) five-mass-chain system, (c) two-link manipulator system used in nicodemus_physicsinformed_2022.
• Figure 3: Nonlinear mass-spring-damper system learning; Reduction of validation errors over time. The DD-PINN training time is significantly lower.
• Figure 4: Nonlinear mass-spring-damper system self-loop prediction of a test path for selected PINN and DD-PINN models. Both models exhibit high prediction accuracy.
• Figure 5: Parameter study of the nonlinear mass-spring-damper system learning results; Scaled MSE to ground truth of $1$ s self-loop prediction path over relative prediction time. DD-PINN models achieve significantly higher prediction accuracy for slightly slower prediction time.