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.