Physics-Informed Echo State Networks for Modeling Controllable Dynamical Systems

TL;DR

PI-ESN addresses the need for controllable, data-efficient dynamic modeling by integrating physical laws into ESN training and allowing external control inputs. The authors introduce PI-ESN with external input (PI-ESN-i) and a self-adaptive balancing loss (PI-ESN-a) that jointly optimize data fit and physics consistency in a discrete-time reservoir setting, making it MPC-ready. Across the Van der Pol oscillator, four-tank system, and ESP-DAE, PI-ESN-a achieves up to 92% relative reduction in test error compared to a plain ESN, with robust performance under parameter uncertainties and improved MPC tracking (IAE reduction from 423.39 to 122.68). The work demonstrates that physics-informed regularization can dramatically improve generalization and control performance for reservoir computing models in limited-data regimes.

Abstract

Echo State Networks (ESNs) are recurrent neural networks usually employed for modeling nonlinear dynamic systems with relatively ease of training. By incorporating physical laws into the training of ESNs, Physics-Informed ESNs (PI-ESNs) were proposed initially to model chaotic dynamic systems without external inputs. They require less data for training since Ordinary Differential Equations (ODEs) of the considered system help to regularize the ESN. In this work, the PI-ESN is extended with external inputs to model controllable nonlinear dynamic systems. Additionally, an existing self-adaptive balancing loss method is employed to balance the contributions of the residual regression term and the physics-informed loss term in the total loss function. The experiments with two nonlinear systems modeled by ODEs, the Van der Pol oscillator and the four-tank system, and with one differential-algebraic (DAE) system, an electric submersible pump, revealed that the proposed PI-ESN outperforms the conventional ESN, especially in scenarios with limited data availability, showing that PI-ESNs can regularize an ESN model with external inputs previously trained on just a few datapoints, reducing its overfitting and improving its generalization error (up to 92% relative reduction in the test error). Further experiments demonstrated that the proposed PI-ESN is robust to parametric uncertainties in the ODE equations and that model predictive control using PI-ESN outperforms the one using plain ESN, particularly when training data is scarce.

Figures (19)

• Figure 1: Echo State Network (ESN) architecture. The reservoir is a recurrent neural network with inner weights and incoming weight connections (solid lines) that are all randomly generated and fixed, projecting the input to a high-dimensional dynamic nonlinear space. A linear readout output layer linearly projects the reservoir states to the desired output (dotted lines). The output $\mathbf{ y}$ can be fed back to the reservoir via $\mathbf{W}^{fb}$.
• Figure 2: Physics-Informed Echo State Network with external Inputs (PI-ESN-i). While PINNs accept explicitly continuous time $t$ as input, PI-ESN-i treats time in a discrete way and implicitly by the recurrent reservoir updates. The ESN output is used to calculate the physics-informed loss function $J_{phy}$ using collocation points and the data loss function $J_{data}$ using data points. These loss functions are scaled by $\lambda_{phy}$ and $\lambda_{data}$, respectively, to form the total loss $J$. The derivative of the loss ${\partial J}/{\partial \mathbf{W}^{out}}$ is calculated by automatic differentiation and used to update the values of the output layer $\mathbf{W}^{out}$.
• Figure 3: Representation of the PI-ESN-i samples used to calculate the total loss function. In solid black, we have the input data $\mathbf{u}[n]$, which can be multidimensional. The desired output is given by the dashed green line, while the ESN's output prediction is given by the dashed pink line. The first section (training datapoints, $t \in [0, T]$) is used in pretraining the ESN's output layer and also in the data loss term included in the total loss function. The second section (collocation points, $t \in [T, T_f]$) is used in the physics-informed loss term included in the total loss function, and training only uses the output prediction $\mathbf{y}[n]$ to compute the loss \ref{['esn:eq:custofisico']}, as indicated in the blue box. The target $\mathbf{\hat{y}}[n]$ in light green is assumed to be unknown for $t > T$, but it is shown to illustrate that $\mathbf{y}[n]$ and $\mathbf{\hat{y}}[n]$ mismatch prior to training, and a performance measurement can be computed after training if it is available.
• Figure 4: Physics-Informed Echo State Network with external Inputs and Self-Adaptive Balancing Loss (PI-ESN-a). The loss function of PI-ESN-a has adaptive scaling parameters $\epsilon_d$ and $\epsilon_f$ that dynamically balance the contributions of $J_{data}$ and $J_{phy}$ into the total loss function (\ref{['esn:eq:gaussianlikelihood_2']}), respectively. An optimizer is employed to minimize this function and update $\mathbf{W}^{out}$ and the adaptive parameters $\epsilon_d$ and $\epsilon_f$. This optimization procedure is outlined in \ref{['algor:adaptive_PI_ESN']}.
• Figure 5: Van der Pol oscillator system with initial conditions $h_1(0) = h_2(0) = 0.1$ shows the variation of oscillation depending on the damping parameter ($\mu$).