Physics-Informed Neural Nets for Control of Dynamical Systems

TL;DR

This paper addresses the challenge of using physics-informed neural networks (PINNs) for control of dynamical systems by introducing a continuous-time PINN extension that accepts control inputs and supports long-range simulations. The authors propose Physics-Informed Neural Nets for Control (PINC), which augments PINNs with initial-state and control-input conditioning and preserves the physics residuals, enabling a self-loop, interval-based prediction within Model Predictive Control (MPC). Experimental results on the Van der Pol oscillator and a four-tank system show that PINC can achieve MPC-ready predictions comparable to RK-based models while offering faster inference over long horizons, and can maintain stability under moderate perturbations. The work highlights sample efficiency, potential scalability to PDE/DAE settings, and practical implications for real-time control where traditional PINNs struggle with horizon length and input variability.

Abstract

Physics-informed neural networks (PINNs) impose known physical laws into the learning of deep neural networks, making sure they respect the physics of the process while decreasing the demand of labeled data. For systems represented by Ordinary Differential Equations (ODEs), the conventional PINN has a continuous time input variable and outputs the solution of the corresponding ODE. In their original form, PINNs do not allow control inputs, neither can they simulate for variable long-range intervals without serious degradation in their predictions. In this context, this work presents a new framework called Physics-Informed Neural Nets for Control (PINC), which proposes a novel PINN-based architecture that is amenable to control problems and able to simulate for longer-range time horizons that are not fixed beforehand, making it a very flexible framework when compared to traditional PINNs. Furthermore, this long-range time simulation of differential equations is faster than numerical methods since it relies only on signal propagation through the network, making it less computationally costly and, thus, a better alternative for simulation of models in Model Predictive Control. We showcase our proposal in the control of two nonlinear dynamic systems: the Van der Pol oscillator and the four-tank system.

Figures (16)

• Figure 1: Representation of the output prediction at a time instant $t_k$, where the proposed actions generate a predicted behavior that reduces the distance between the value predicted by the model and a reference trajectory.
• Figure 2: The PINC network has initial state $\mathbf{y}(0)$ of the dynamic system and control input $\mathbf{u}$ as inputs, in addition to continuous time scalar $t$. Both $\mathbf{y}(0)$ and $\mathbf{u}$ can be multidimensional. The output $\mathbf{y}(t)$ corresponds to the state of the dynamic system as a function of $t \in [0,T]$, and initial conditions given by $\mathbf{y}(0)$ and $\mathbf{u}$. The deep network is fully connected even though not all connections are shown.
• Figure 3: Modes of operation of the PINC network. (a) PINC net operates in self-loop mode, using its own output prediction as next initial state, after $T$ seconds. This operation mode is used within one iteration of MPC, for trajectory generation until the prediction horizon of MPC completes (predicted output from Fig. \ref{['fig:mpc_pred']}). (b) Block diagram for PINC connected to the plant. One pass through the diagram arrows corresponds to one MPC iteration applying a control input $\mathbf{u}$ for $T_s$ timesteps for both plant and PINC network. Note that the initial state of the PINC net is set to the real output of the plant. In practice, in MPC, these two operation modes are executed in an alternated way (optimization in the prediction horizon, and application of control action).
• Figure 4: Representation of a trained PINC network evolving through time in self-loop mode (Fig. \ref{['fig:pinc_feedback']}) for trajectory generation in prediction horizon. The top dashed black curve corresponds to a predicted trajectory $\mathbf{y}$ of a hypothetical dynamic system in continuous time. The states $\mathbf{y}[k]$ are snapshots of the system in discrete time $k$ positioned at the equidistant vertical lines. Between two vertical lines (during the inner continuous interval between steps $k$ and $k+1$), the PINC net learns the solution of an ODE with $t \in [0,T]$, conditioned on a fixed control input $\mathbf{u}[k]$ (blue solid line) and initial state $\mathbf{y}(0)$ (green thick dashed line). Control action $\mathbf{u}[k]$ is changed at the vertical lines and kept fixed for $T$ seconds, and the initial state $\mathbf{y}(0)$ in the interval between steps $k$ and $k+1$ is updated to the last state of the previous interval $k-1$ (indicated by the red curved arrow). The PINC net can directly predict the states at the vertical lines without the need to infer intermediate states $t < T$ as numerical simulation does. Here, we assume that $T=T_s$ and, thus, the number of discrete timesteps $M$ is equal to the length of the prediction horizon in MPC.
• Figure 5: Analysis of the PINC net for the Van der Pol Oscillator. The network training time is fixed to a constant number of iterations. The MSE validation error is computed according to Equation (\ref{['eq:mse_gen']}). (a) The $\log_{10}$ of the MSE error as a function of network complexity, averaged over 10 different simulations. The best generalization error ($10^{-2.87}$) is achieved with a deep network of 10 layers with 20 neurons each. (b) The effect of the number of collocation points $N_f$ and data points $N_t$ on generalization performance, averaged over 5 different randomly initialized networks.