Solving parametric PDE problems with artificial neural networks
For researchers in computational science and engineering, this work offers a practical method to handle high-dimensional parametric PDEs, though it is incremental as it builds on existing neural-network PDE solvers.
The paper proposes a neural-network-based method to solve parametric PDEs by parameterizing the physical quantity of interest as a function of input coefficients, addressing the curse of dimensionality. The approach is demonstrated to be simple and accurate on notable PDE examples from engineering and physics.
The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of physical quantities derived from a PDE can be captured by a few features on the space of the coefficient fields. Based on such an observation, we propose using a neural-network (NN) based method to parameterize the physical quantity of interest as a function of input coefficients. The representability of such quantity using a neural-network can be justified by viewing the neural-network as performing time evolution to find the solutions to the PDE. We further demonstrate the simplicity and accuracy of the approach through notable examples of PDEs in engineering and physics.