# Physics Enhanced Deep Surrogates for the Phonon Boltzmann Transport Equation

arXiv: https://arxiv.org/abs/2512.05976 · HTML: https://mlsanity.com/paper/2512.05976/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 94 references that are in the map.
- First posted on arXiv 25 Nov 2025.

## Builds on (most cited first, 24 of 24)

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- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Advances and Challenges in Meta-Learning: A Technical Review](https://mlsanity.com/paper/2307.04722/) (2023)
- [Physics Informed Neural Networks for Simulating Radiative Transfer](https://mlsanity.com/paper/2009.13291/) (2020)
- [Active learning of deep surrogates for PDEs: Application to metasurface design](https://mlsanity.com/paper/2008.12649/) (2020)
- [Physics-enhanced deep surrogates for partial differential equations](https://mlsanity.com/paper/2111.05841/) (2021)
- [Deep learning approach based on dimensionality reduction for designing electromagnetic nanostructures](https://mlsanity.com/paper/1902.03865/) (2019)
- [Inverse Design in Nanophotonics via Representation Learning](https://mlsanity.com/paper/2507.00546/) (2025)
- [HiLAB: A Hybrid Inverse-Design Framework](https://mlsanity.com/paper/2505.17491/) (2025)
- [Deep Learning Reveals Underlying Physics of Light-matter Interactions in Nanophotonic Devices](https://mlsanity.com/paper/1905.06889/) (2019)

## Abstract (arXiv)

Designing materials with controlled heat flow at the nano-scale is central to advances in microelectronics, thermoelectrics, and energy-conversion technologies. At these scales, phonon transport follows the Boltzmann Transport Equation (BTE), which captures non-diffusive (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. 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 nano-scale behavior. PEDS reduces training-data requirements by up to 70% compared with purely data-driven baselines, achieves roughly 5% fractional error with only 300 high-fidelity BTE simulations, and enables efficient design of porous geometries spanning 12-85 W m-1 K-1 with average design errors of 4%. The learned mixing parameter recovers the ballistic-diffusive transition and improves out of distribution robustness. These results show that embedding simple, differentiable low-fidelity physics can dramatically increase surrogate data-efficiency and interpretability, making repeated PDE-constrained optimization practical for nano-scale thermal-materials design.