6.4. Physics-informed neural networks
Can we use prior physics-based knowledge to avoid overfitting or non-physical predictions? From a conceptual point of view, can we supplement ML with a set of known physics-based equations, an approach that drives MSM models in engineering disciplines? While data-driven methods can provide solutions that are not constrained by preconceived notions or models, their predictions should not violate the fundamental laws of physics. There are well-known examples of deep learning neural networks that appear to be highly accurate but make highly inaccurate predictions when faced with data outside their training regime, and others that make highly inaccurate predictions based on seemingly minor changes to the target data [147]. To address this ubiquitous issue of purely ML-based approaches, numerous opportunities to combine machine learning and multiscale modeling towards a priori satisfying the fundamental laws of physics, and, at the same time, preventing overfitting of the data.
A potential solution is to combine deterministic and stochastic models. Coupling the deterministic governing equations MSM models — the balance of mass, momentum, and energy — with the stochastic equations of systems biology and biophysical systems — cell-signaling networks or reaction-diffusion equations — could help guide the design of computational models for otherwise ill-posed problems. Physics-informed neural networks (PINN) [148] is a promising approach that employs deep neural networks and leverages their well-known capability as universal function approximators [149]. In this setting, we can directly tackle nonlinear problems without the need for committing to any prior assumptions, linearization, or local time-stepping. PINNs exploit recent developments in automatic differentiation [150] to differentiate neural networks concerning their input coordinates and model parameters to obtain physics informed neural networks. Such neural networks are constrained to respect any symmetry, invariance, or conservation principles originating from the physical laws that govern the observed data, as modeled by general time-dependent and nonlinear partial differential equations. This construction allows us to tackle a wide range of problems in computational science and introduces a potentially disruptive technology leading to the development of new data-efficient and physics-informed learning machines, new classes of numerical solvers for partial differential equations, as well as new data-driven approaches for model inversion and systems identification.