# A hybrid analytical-PINN model for subsurface simulation of geothermal heat exchangers in heterogeneous underground

arXiv: https://arxiv.org/abs/2607.12271 · HTML: https://mlsanity.com/paper/2607.12271/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 31 references that are in the map.
- First posted on arXiv 14 Jul 2026.

## Builds on (most cited first, 8 of 8)

- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [Failure-informed adaptive sampling for PINNs, Part II: combining with re-sampling and subset simulation](https://mlsanity.com/paper/2302.01529/) (2023)
- [A Shallow Ritz Method for Elliptic Problems with Singular Sources](https://mlsanity.com/paper/2107.12013/) (2021)
- [Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method](https://mlsanity.com/paper/2209.02931/) (2022)
- [Physics-Informed DeepONet Coupled with FEM for Convective Transport in Porous Media with Sharp Gaussian Sources](https://mlsanity.com/paper/2508.19847/) (2025)
- [Optimization approaches for the design and operation of open-loop shallow geothermal systems](https://mlsanity.com/paper/2307.11244/) (2023)
- [Spatial analysis of thermal groundwater use based on optimal sizing and placement of well doublets](https://mlsanity.com/paper/2311.01816/) (2023)
- [Optimization of Closed-Loop Shallow Geothermal Systems Using Analytical Models](https://mlsanity.com/paper/2603.24957/) (2026)

## Abstract (arXiv)

Accurate and efficient prediction of subsurface temperature fields is essential for the design and operation of borehole heat exchanger (BHE) systems. Here we develop a parametric hybrid analytical and physics-informed neural network (PINN) framework for long-term multi-BHE simulations in heterogeneous underground. The method analytically extracts the singular line source response and enables the effective training of neural correction associated with subsurface heterogeneity. An explicit parametrization of the thermal conductivity allows physics-informed learning of a single feedforward neural network to generalize across different subsurface conditions. By formulating the correction in borehole-centered relative coordinates, the learned correction can be reused as a universal corrector through spatial and temporal superposition principles. Numerical experiments based on the infinite line source (ILS), finite line source (FLS) and moving finite line source (MFLS) models show that the hybrid method outperforms analytical approximations with stable accuracy over long simulation horizons and achieves orders-of-magnitude speedups over traditional solvers. The proposed framework therefore combines the efficiency of analytical models with the ability of numerical methods to capture heterogeneous subsurface physics, providing a fast and accurate approach for repeated long-term simulation of multi-BHE systems.