Physics-Enhanced Deep Surrogate for the Phonon Boltzmann Transport Equation

Abstract

Abstract. Designing materials with controlled heat flow at the nanoscale is central to advances in micro-electronics, thermoelectrics, and energy-conversion technologies. At these scales, phonon transport follows the Boltzmann Transport Equation (BTE), which captures nondiffusive (ballistic) effects but is too costly to solve repeatedly in inverse-design loops. Existing surrogate approaches trade speed for accuracy: fast macroscopic solvers can overestimate conductivities by hundreds of percent, while recent data-driven operator learners often require thousands of high-fidelity simulations. This creates a need for a fast, data-efficient surrogate that remains reliable across ballistic and diffusive regimes. We introduce a physics-enhanced deep surrogate (PEDS) that combines a differentiable Fourier solver with a neural generator and couples it with uncertainty-driven active learning (AL). The Fourier solver acts as a physical inductive bias, while the network learns geometry-dependent corrections and a mixing coefficient that interpolates between macroscopic and nanoscale behavior. PEDS reduces training-data requirements by up to 70% compared with purely data-driven baselines, achieves roughly 5% fractional error (FE) with only 300 high-fidelity BTE simulations, and enables efficient design of porous geometries spanning 12–85 W m−1K−1 with average design errors of 4%. The learned mixing parameter recovers the ballistic–diffusive transition and improves the out-of-distribution robustness. These results show that embedding simple, differentiable low-fidelity physics dramatically increases the surrogate data-efficiency and interpretability, making repeated PDE-constrained optimization practical for nanoscale thermal-materials design.