Using Neural Networks to Solve the Inverse Problem of Heat Conduction in Complex Bodies

This paper presents an integrated computational methodology for solving the inverse heat conduction problem in geometrically complex, anisotropic reactor fuel elements. The methodology couples a parallel cell-centered finite volume forward solver capable of treating fully anisotropic thermal conductivity tensors, mixed boundary conditions of all three classical kinds, and radiation effects on unstructured three-dimensional meshes with a multilayer perceptron (MLP) trained on large datasets of numerically generated forward solutions and optimized by the Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) quasi-Newton method. The forward solver operates in parallel using MPI with recursive coordinate bisection domain decomposition and a Jacobi-preconditioned conjugate gradient linear solver. Training datasets are generated by repeated solution of the steady-state heat conduction equation over randomly sampled physical parameters, and surface temperatures at up to 20 probe points serve as network inputs for recovery of the unknown thermophysical parameters. The approach is validated on twisted-geometry fuel rod configurations discretized on an unstructured mesh of 73 008 cells. For single-parameter inverse problems—recovery of thermal conductivity, heat transfer coefficient, or volumetric heat generation rate—the trained MLP achieves maximum prediction errors below 0.034%, with a minimal two–hidden layer network (8, 8 neurons) proving entirely sufficient. For the combined simultaneous recovery of the above parameters from 20 surface measurements, fixing at least one boundary condition resolves the intrinsic ill-posedness; a three-layer network (16, 32, 16 neurons) reduces maximum errors to below 0.02%. The methodology is further extended to anisotropic materials: The diagonal conductivity tensor components are recovered with a maximum error of 0.15%, and all six independent components of the fully general symmetric tensor are reconstructed with satisfactory accuracy for the dominant diagonal entries. A practical rule established by the numerical experiments is that the number of probe points should be approximately 4 to 5 times the number of parameters to be recovered simultaneously. Once the model is trained, inference requires only a single forward pass through the network (O(P) operations), making the approach directly applicable to real-time diagnostics, digital-twin frameworks, and in-reactor monitoring scenarios.