Fractional Physics Informed Neural Networks for Surrogate Modeling of Non-Markovian Discrete-Time Quantum Walks.

Predicting anomalous diffusion in quantum walks with non-Markovian environmental noise is computationally demanding. We introduce FracPINN, a fractional physics-informed neural network that embeds a fully differentiable, PyTorch-based Caputo PDE solver into a classical LSTM encoder. Rather than replacing classical predictors, FracPINN acts as a compact physics regularizer that constrains the inference with emergent fractional transport dynamics; every physics-informed loss component is strictly label-free, and the ground-truth exponent enters only through an explicitly supervised regression term. Evaluated on 3709 non-Markovian DTQW simulations with exponentially correlated Gaussian coin noise (filtered from 5000 raw samples to the physically admissible exponent range), FracPINN achieves a mean absolute error of 0.214 and R2=0.682 on held-out test data, outperforming classical baselines by 5.3% in terms of the MAE overall, while adding only four interpretable physical parameters. Notably, gains concentrate in the sub-diffusive regime where memory effects dominate, with a 13.5% MAE improvement over the baseline there, yet the normal and super diffusive accuracy remains intact. Once trained, the surrogate reduces inference from seconds of simulation to fractions of a millisecond per sample. These results show that our fractional PDE networks are most compelling as targeted physics refinements with minimal overhead within stable classical pipelines.