Variational formulation of physics-informed neural networks

Physics-informed neural networks (PINNs) are deep neural networks solving differential equations by minimizing a phenomenological loss function formed from these equations. The unknown variable is iteratively computed at numerous points in the domain using backpropagation and hence providing a mesh-free approach to compute solutions. However, the involvement of higher-order derivatives in equations describing many physical systems results in expensive computational costs. Additionally, while solving coupled differential equations, PINNs introduce more complexity due to ad hoc weight factors that are either determined manually or via complex algorithms. We propose a variational PINN (vPINN) algorithm that optimizes functionals in integral form (e.g., Lagrangian, Hamiltonian, or Rayleighian) to overcome the aforementioned disadvantages: vPINN naturally involves lower-order derivatives and replaces the ad hoc weight factors with rigorous physical scales. Our preliminary simulations show promising agreement between vPINN and known solutions for benchmark systems such as the Poisson's equation and other ordinary differential equations. More complex equation sets, including elliptic and nonlinear partial differential equations, are under further investigation.