Detecting Unrepresented Physics in Hybrid Machine Learning Surrogates of Geothermal Systems using Kolmogorov-Arnold Networks

Accurate and efficient modelling of geothermal reservoirs is important for sustainable energy production and for the reliable assessment of operational risks. Predicting thermo-hydraulic (TH) system evolution under varying injection and production scenarios remains computationally challenging, particularly when physical knowledge of the subsurface system is incomplete and observational data are sparse. High-fidelity finite-element simulators are typically used to provide physics-based predictions of coupled flow and heat transport governed by complex partial-differential equations (PDEs). Such full-order simulations are, however, often prohibitively expensive for real-time forecasting, which is essential, for instance, in the context of digital twins.

Physics-based machine-learning (PBML) approaches, such as the non-intrusive reduced basis (NIRB) method address this challenge by constructing physics-consistent surrogate models that project full-order simulation outputs onto a low-dimensional subspace learned from representative snapshots. By retaining only the dominantbasis functions, the NIRB surrogate enables orders-of-magnitude speedup in parametric predictions while staying consistent with the physical transport mechanisms and structural assumptions on fracture networks encoded in the full-order model. Despite these advantages, classical NIRB surrogates are intrinsically limited to the physical regimes represented by the governing PDEs, and consequently by the training simulations. If the surrogate does not fully capture the observed system behaviour, it is important to detect and adapt to missing or misrepresented local physics revealed by observational data, such as unmodeled convective heat transport or flow channelling arising from fracture activation.

To address this need, we propose a complementary residual-learning framework that augments a baseline NIRB surrogate with parameter-to-state maps of residual temperature and pressure fields learned by Kolmogorov-Arnold Networks (KANs). The residual, defined as the difference between observed data (or a synthetic reference solution) and the NIRB model prediction, is interpreted as a proxy for missing or misrepresented physics not explicitly captured by the baseline model. KANs represent mappings as sums of learned univariate functions and provide explicit access to the functional structure of parameter dependence. Thereby, KANs could act as interpretable discrepancy models by learning the residual between observations and NIRB predictions. By analysing the dominant functional families emerging in the learned residual, such as linear dependence characteristic of conduction-dominated regimes or exponential dependence associated with convection, KANs can provide diagnostic insight into missing thermo-hydraulic processes and their relevance across parameter regimes.

We validate the proposed approach synthetically by comparing a conduction-only NIRB surrogate against synthetic reference observations generated with an advection–diffusion model. We expect that KAN-based residual learning both improves predictive accuracy and reveals clear functional signatures of missing convective physics, even when only pointwise information is available. As an outlook, we aim to apply this workflow to real geothermal case studies, where sparse temperature and pressure measurements are available at well locations. In such settings, functional-family learning of residuals offers a promising pathway to improve surrogate predictions and to enhance the physical interpretability of geothermal systems, ultimately supporting more reliable assessments of reservoir behaviour.