A Physics Informed Neural Network (PINN) Methodology for Coupled Moving Boundary PDEs

TL;DR

The paper tackles solving coupled moving-boundary PDEs in binary alloy solidification with a physics-informed neural network (PINN). It introduces a three-network architecture (for $ heta$, $C_l$, and $oldsymbol{ extepsilon^*}$) plus a trainable $C_s$, and employs a novel sequential training regime that alternates causal temporal learning with adaptive loss weighting to progressively constrain the solution. The results demonstrate that the combined approach captures the interface dynamics and discontinuous composition with high fidelity, aligning closely with analytical benchmarks and outperforming single-strategy PINN implementations. The framework is particularly suited to low-data regimes and generalizes to other transient multiphysics problems involving moving interfaces and discontinuities.

Abstract

Physics-Informed Neural Network (PINN) is a novel multi-task learning framework useful for solving physical problems modeled using differential equations (DEs) by integrating the knowledge of physics and known constraints into the components of deep learning. A large class of physical problems in materials science and mechanics involve moving boundaries, where interface flux balance conditions are to be satisfied while solving DEs. Examples of such systems include free surface flows, shock propagation, solidification of pure and alloy systems etc. While recent research works have explored applicability of PINNs for an uncoupled system (such as solidification of pure system), the present work reports a PINN-based approach to solve coupled systems involving multiple governing parameters (energy and species, along with multiple interface balance equations). This methodology employs an architecture consisting of a separate network for each variable with a separate treatment of each phase, a training strategy which alternates between temporal learning and adaptive loss weighting, and a scheme which progressively reduces the optimisation space. While solving the benchmark problem of binary alloy solidification, it is distinctly successful at capturing the complex composition profile, which has a characteristic discontinuity at the interface and the resulting predictions align well with the analytical solutions. The procedure can be generalised for solving other transient multiphysics problems especially in the low-data regime and in cases where measurements can reveal new physics.

Figures (9)

• Figure 1: (a) Indicative thermal and compositional profiles in a one-dimensional system undergoing solidification of a superheated alloy melt. The left side boundary is subjected to cooling. The two phases and the sharp boundary separating them are shown at an intermediate stage. (b) the associated phase diagram showing equilibrium phase transition lines
• Figure 2: Schematic representations of the architectures of the 3 neural networks for (a) temperature distribution in both phases, (b) liquid concentration, and (c) inteface position
• Figure 3: Schematic representation of the physics-informed component of the architecture for solving the solidification problem showing the make up of loss terms and the optimisation loops
• Figure 4: Results of the proposed model showing (a) Loss curve, and (b) interface movement after first iteration of causal training
• Figure 5: (a) Predicted temperature distribution $\theta(x)$ showing a jump in the gradient at the interface, and (b) composition $C(x)$, compared with the analytical solution at $t=1.725$ s