Quadratic Residual Networks: A New Class of Neural Networks for Solving Forward and Inverse Problems in Physics Involving PDEs

We propose quadratic residual networks (QRes) as a new type of parameter-efficient neural network architecture, by adding a quadratic residual term to the weighted sum of inputs before applying activation functions. With sufficiently high functional capacity (or expressive power), we show that it is especially powerful for solving forward and inverse physics problems involving partial differential equations (PDEs). Using tools from algebraic geometry, we theoretically demonstrate that, in contrast to plain neural networks, QRes shows better parameter efficiency in terms of network width and depth thanks to higher non-linearity in every neuron. Finally, we empirically show that QRes shows faster convergence speed in terms of number of training epochs especially in learning complex patterns.
Comment: Accepted by SIAM International Conference on Data Mining (SDM21)