Physics-Informed Deep Learning for Industrial Processes: Time-Discrete VPINNs for heat conduction

TL;DR

A Variational Physics-Informed Neural Network (VPINN) designed for parabolic problems is presented, which combines a classical time discretization with a composed loss function, which minimizes the residual's dual norm at every time step.

Abstract

Neural networks offer powerful tools to solve partial differential equations (PDEs). We present a Variational Physics-Informed Neural Network (VPINN) designed for parabolic problems. Our approach combines a classical time discretization with a composed loss function, which minimizes the residual's dual norm at every time step. We validate the framework by modeling the freezing of coffee extracts in an industrial cylinder. The simulation accounts for temperature-dependent properties and experimental data. It successfully captures the thermal dynamics of the process.

Figures (7)

• Figure 1: Neural network architecture
• Figure 2: Comparison of the neural network approximation ($u^n_{\theta}$) and the exact solution at four different time steps ($n=1, 32, 64, 128$).
• Figure 3: Evolution of the loss function during training (left). The relative errors ($L^2(\Omega)$ and $H^1_0(\Omega)$) (right).
• Figure 4: Thermophysical properties of the coffee extract in terms of the temperature.
• Figure 5: Temporal evolution of the temperature profile. The non-linear solution $u$ (left) incorporates empirical thermophysical parameters, while the linear control solution $u_0$ (right) assumes constant properties. Data-driven boundary values are marked with $\times$, and the midpoint temperature is highlighted with $\bullet$.

Theorems & Definitions (2)

• Remark 1
• Remark 2