Machine learning for parameter estimation

Abstract

Mathematical modeling provides the critical infrastructure for the quantitative analysis of any discipline. Models are diverse in mathematical sophistication and scope, objectives, and philosophical viewpoint (statistical, dynamical, game-theoretic, etc.). Often generated from expert knowledge, first principles, and/or experimental observations, models require parametrization by construction (1). Physics has had a long history of parameter estimation leading to the discovery of fundamental constants (gravity, the speed of light, Planck’s constant, etc.) and dimensionless quantities through the Buckingham Pi theorem (Reynolds number, Fresnel number, Mach number, etc.) (2). The modeling of modern complex systems, however, makes parameter estimation exceptionally challenging since, unlike the physics examples given, there are typically a large number of parameters that need to be simultaneously estimated. Machine learning (ML) algorithms are now allowing for flexible, fast, and robust optimization architectures for solving the largescale inverse problems associated with parameter estimation in complex systems (3). In a recent transformative use of ML, Gaskin et al. (4) have used the power of neural differential equations (5, 6) to generate a computationally simple and fast method for retrieving accurate probability densities for model parameters in difficult-to-model agent-based models (ABMs) which act as forward solvers for systems of ordinary, partial, and/or stochastic differential equations. The algorithm advocated uses a neural network for computing a gradient for updating parameter values, thus providing an efficient and targeted use of ML which is interpretable in the parameter estimation problem. This helps advance the use of data-driven methods in challenging application areas. Moreover, the algorithm harmoniously blends traditional scientific computing of the underlying complex dynamics with machine learning for the constrained goal of learning the best parameters of the simulations. Machine learning algorithms are especially important to consider in the arena of high-dimensional complex systems where traditional methods of parameter estimation struggle due to computational intractability, uncertainty, and/or inaccuracy. Application areas that are particularly relevant include systems characterized by agent-based models (ABMs) (7) which consider high-dimensional models of agents (entities or nodes) that typically interact through differential and stochastic differential equation-based rules. By construction, such a modeling paradigm generates a large number of parameters that must be estimated in order to produce dynamics consistent with observational data. Alternatively, one can model complex systems using continuous time and space partial differential equations for the underlying spatio-temporal dynamics. Again, the inverse problem associated with this high-dimensional modeling paradigm also poses many difficulties in reconstructing accurate estimates of the PDE parameters required to produce dynamics consistent with observation data (8). In both of these modeling paradigms, estimating a large number of parameters for the underlying dynamical model is ill-posed and requires regularization. Markov-Chain Monte Carlo (MCMC) methods,