Physics-informed State-space Neural Networks for Transport Phenomena

TL;DR

The paper addresses the need for physically consistent models in transport-dominated systems by introducing Physics-informed State-space Neural Networks (PSMs) that fuse sensor data with conservation laws via PDE residuals within a discrete-time, end-to-end differentiable state-space framework. By extending PINNs with a continuous control-input and initial-condition aware architecture, PSMs deliver accurate forward dynamics across the spatial domain and support multitask applications such as model-based control and physics-based diagnostics. Validation on heated-channel and cooling-loop in silico experiments shows PSMs outperform purely data-driven ANNs, especially for temperature predictions and in noisy conditions, while enabling constraint enforcement and fault detection through PDE residual analysis. The results suggest PSMs as a viable foundation for digital twins and real-time operation in transport-dominated engineering systems, with implications for control, monitoring, and autonomous optimization.

Abstract

This work introduces Physics-informed State-space neural network Models (PSMs), a novel solution to achieving real-time optimization, flexibility, and fault tolerance in autonomous systems, particularly in transport-dominated systems such as chemical, biomedical, and power plants. Traditional data-driven methods fall short due to a lack of physical constraints like mass conservation; PSMs address this issue by training deep neural networks with sensor data and physics-informing using components' Partial Differential Equations (PDEs), resulting in a physics-constrained, end-to-end differentiable forward dynamics model. Through two in silico experiments -- a heated channel and a cooling system loop -- we demonstrate that PSMs offer a more accurate approach than a purely data-driven model. In the former experiment, PSMs demonstrated significantly lower average root-mean-square errors across test datasets compared to a purely data-driven neural network, with reductions of 44 %, 48 %, and 94 % in predicting pressure, velocity, and temperature, respectively. Beyond accuracy, PSMs demonstrate a compelling multitask capability, making them highly versatile. In this work, we showcase two: supervisory control of a nonlinear system through a sequentially updated state-space representation and the proposal of a diagnostic algorithm using residuals from each of the PDEs. The former demonstrates PSMs' ability to handle constant and time-dependent constraints, while the latter illustrates their value in system diagnostics and fault detection. We further posit that PSMs could serve as a foundation for Digital Twins, constantly updated digital representations of physical systems.

Figures (13)

• Figure 1: A visualization of the nomenclature and their relationships. The red lines indicate variables needed at each step $k$, i.e., the initial condition and input, $x_{0,k}$, and $v_k$. The blue arrows indicate the prediction of the next state, $x_k$, by the numerical solver or PSM. The dashed arrows indicate the aliasing of $x_k$ as $x_{0,k+1}$.
• Figure 2: PSM architecture and training workflow. Top: The neural network architecture displaying relative sizes of the MLP layers. Bottom: Measurement and physics-informed losses use output from $\mathcal{F}\left(\mathcal{X}_m\right)$ and $\mathcal{F}\left(\mathcal{X}_p\right)$, respectively. The combined losses are used to update the parameters of the neural network, $\theta$.
• Figure 3: Configuration of the pipes in series. Left: Layout of the pipes and identifications of the locations of imposed BCs and heat source term. Right: A list of the fixed outlet pressure BC, and ranges for the inlet BCs.
• Figure 4: An overview of the training dataset for the heated channel configuration. Left: Velocity and temperature BCs at the inlet that are manipulated in time. The experiment chosen to display the solution is annotated ("Exp. 8"). Right: Numerical solution of all field variables over the entire spatiotemporal domain.
• Figure 5: Contrasting the performance of PSM and ANN models in predicting the evolution of pressure, velocity, and temperature for the heated channel. Subplots A-B: control input setting from the test dataset. Subplots C-K: field temporal evolution at fixed positions. Subplots L-T: field spatial distribution at fixed times.