Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm
This provides a computationally efficient method for solving forward and inverse PDE problems in scientific computing, though it is incremental as it builds on existing PINN frameworks with a different optimization approach.
The paper tackles solving nonlinear PDEs with shallow physics-informed neural networks (PINNs) by using the Levenberg-Marquardt algorithm for optimization, showing it outperforms BFGS in convergence speed, accuracy, and loss values on benchmark problems like Burgers and Schrödinger equations.
This work investigates the use of shallow physics-informed neural networks (PINNs) for solving forward and inverse problems of nonlinear partial differential equations (PDEs). By reformulating PINNs as nonlinear systems, the Levenberg-Marquardt (LM) algorithm is employed to efficiently optimize the network parameters. Analytical expressions for the neural network derivatives with respect to the input variables are derived, enabling accurate and efficient computation of the Jacobian matrix required by LM. The proposed approach is tested on several benchmark problems, including the Burgers, Schrödinger, Allen-Cahn, and three-dimensional Bratu equations. Numerical results demonstrate that LM significantly outperforms BFGS in terms of convergence speed, accuracy, and final loss values, even when using shallow network architectures with only two hidden layers. These findings indicate that, for a wide class of PDEs, shallow PINNs combined with efficient second-order optimization methods can provide accurate and computationally efficient solutions for both forward and inverse problems.