Can we infer probability distributions from cumulants? Probabilistic approaches to inverse problems

Inverse problems, which involve estimating system inputs from outputs, are prevalent across science and engineering. Their ill-posed nature often makes finding numerically stable and unique solutions challenging. This seminar explores probabilistic methods for reconstructing distributions from a finite set of their moments or cumulants. We apply the Maximum Entropy Method (MEM) and Gaussian Process (GP) to reconstruct net-baryon number distributions across the QCD chiral crossover region using cumulant data from the STAR experiment and functional renormalization group (fRG) calculations. Our results demonstrate how higher-order cumulants shape distribution tails, while anomalous features in the reconstructed distributions provide constraints on the input cumulants. We also discuss deep learning approaches for distribution reconstruction from cumulants and present our recent work on physics-informed neural networks (PINNs) for solving fRG equations.