JAX-Accelerated Neuroevolution of Physics-informed Neural Networks: Benchmarks and Experimental Results

This paper introduces the use of evolutionary algorithms for solving differential equations. The solution is obtained by optimizing a deep neural network whose loss function is defined by the residual terms from the differential equations. Recent studies have used stochastic gradient descent (SGD) variants to train these physics-informed neural networks (PINNs), but these methods can struggle to find accurate solutions due to optimization challenges. When solving differential equations, it is important to find the globally optimum parameters of the network, rather than just finding a solution that works well during training. SGD only searches along a single gradient direction, so it may not be the best approach for training PINNs with their accompanying complex optimization landscapes. In contrast, evolutionary algorithms perform a parallel exploration of different solutions in order to avoid getting stuck in local optima and can potentially find more accurate solutions. However, evolutionary algorithms can be slow, which can make them difficult to use in practice. To address this, we provide a set of five benchmark problems with associated performance metrics and baseline results to support the development of evolutionary algorithms for enhanced PINN training. As a baseline, we evaluate the performance and speed of using the widely adopted Covariance Matrix Adaptation Evolution Strategy (CMA-ES) for solving PINNs. We provide the loss and training time for CMA-ES run on TensorFlow, and CMA-ES and SGD run on JAX (with GPU acceleration) for the five benchmark problems. Our results show that JAX-accelerated evolutionary algorithms, particularly CMA-ES, can be a useful approach for solving differential equations. We hope that our work will support the exploration and development of alternative optimization algorithms for the complex task of optimizing PINNs.