Physics-informed neural network amplitude and phase modeling of steady-state thermal waves in layered structures

A physics-informed neural network (PINN) approach is proposed to simulate steady-state amplitude and phase of thermal waves in multilayered solids with spatial lateral heat loss gradients under Robin boundary conditions. Thermal-wave propagation in heterogeneous structures is modeled via spatially varying thermal conductivity and heat loss coefficient. Using separation of variables, the unsteady diffusion equation is converted into coupled amplitude-phase equations under harmonic steady-state conditions. This conversion removes the explicit time coordinate from the harmonic problem and enables the networks to learn the corresponding spatial amplitude and phase fields directly. A unified parameterization of thermal conductivity enables a single network to handle multiple layers without separate training or explicit interface conditions. The framework is validated on four benchmark cases: a 3D point source in a homogeneous medium, a planar source with uniform heat loss, a case with spatially varying heat loss, and a three-layer composite with stepwise diffusivity and varying heat loss. Root-mean-square errors relative to analytical solutions remain below 10−5. Transfer learning, reusing a pretrained model, reduces training iterations up to threefold. Application to experimental data from a green metal powder composite yields an estimated thermal diffusivity of 6.12 × 10−6 m2/s using only the measured phase profile, agreeing within 1% of an independent reference value. The results demonstrate that PINNs provide accurate forward simulations and enable inverse parameter estimation from experimental measurements. Extensions to higher dimensions, thermoelasticity, nonlinearities, and other inverse problems are feasible.