Predicting nonlinear stochastic and quantum dynamics without PDEs

Physics aims to make quantitative predictions given information about initial and boundary conditions. Predicting the behavior of quantum and nonlinear stochastic systems often involves formulating and solving linear operator equations that have a characteristic eigenbasis. Once this basis has been identified, prediction reduces to a straightforward series evaluation. In many practically relevant situations, however, the natural bases cannot be calculated analytically. Here, we introduce a method that skips the model formulation step of the traditional prediction pathway by directly discovering the fundamental bases from data through an interpretable matrix factorization. Given the basis, predictions can be made by computing a simple combination of matrix products, providing a fast forecasting procedure when no explicit dynamical equation is known.