Adaptive PINN-SP1: NTK-Based Penalties for Noisy Inverse Problems in Radiative Transfer

We consider the inverse problem of estimating the absorption coefficient in a two-dimensional participating medium governed by the $SP_1$ approximation of the particle transport equation. While Physics-Informed Neural Networks (PINNs) are promising for such tasks, estimating a piecewise constant absorption from discrete boundary detectors introduces severe loss imbalances that destabilize training. To overcome this, our methodology relies on adaptive penalties, based on neural tangent kernel theory (NTK), to dynamically weight the loss components, ensuring a stable and robust optimization process. We evaluate this adaptive framework using radiative flux data from discrete boundary detectors with noise. Our results demonstrate that incorporating adaptive penalties successfully mitigates training instabilities, allowing the model to accurately estimate both the location and magnitude of the absorption coefficient even in noisy scenarios.