Data Denoising and Derivative Estimation for Data-Driven Modeling of Nonlinear Dynamical Systems

Data-driven modeling of nonlinear dynamical systems is often hampered by measurement noise. We propose a denoising framework, called Runge-Kutta and Total Variation Based Implicit Neural Representation (RKTV-INR), that represents the state trajectory with an implicit neural representation (INR) fitted directly to noisy observations. Runge-Kutta integration and total variation are imposed as constraints to ensure that the reconstructed state is a trajectory of a dynamical system that remains close to the original data. The trained INR yields a clean, continuous trajectory and provides accurate first-order derivatives via automatic differentiation. These denoised states and derivatives are then supplied to Sparse Identification of Nonlinear Dynamics (SINDy) to recover the governing equations. Experiments demonstrate effective noise suppression, precise derivative estimation, and reliable system identification.

Neural Likelihood Approximation for Integer Valued Time Series Data

Stochastic processes defined on integer valued state spaces are popular within the physical and biological sciences. These models are necessary for capturing the dynamics of small systems where the individual nature of the populations cannot be ignored and stochastic effects are important. The inference of the parameters of such models, from time series data, is difficult due to intractability of the likelihood; current methods, based on simulations of the underlying model, can be so computationally expensive as to be prohibitive. In this paper we construct a neural likelihood approximation for integer valued time series data using causal convolutions, which allows us to evaluate the likelihood of the whole time series in parallel. We demonstrate our method by performing inference on a number of ecological and epidemiological models, showing that we can accurately approximate the true posterior while achieving significant computational speed ups in situations where current methods struggle.