In this talk, I will present a statistical finite element method for nonlinear, time-dependent phenomena. The statistical finite element method (statFEM) is a statistical augmentation of the finite element method that enables model-data synthesis through the admission of model misspecification inside of the governing equations, as represented by a Gaussian process. The method is Bayesian, coherently updates model mismatch upon receipt of observed data, and is applicable to a wide range of problems across science and engineering for which finite element methods are appropriate. Scalability is ensured through making a low-rank approximation to the posterior covariance matrix. I'll first introduce the statFEM, before detailing the methodology for nonlinear problems. I will discuss various case studies, applying the method to experimental and synthetic data in canonical nonlinear PDEs.