Achieving High Accuracy with PINNs via Energy Natural Gradients
This addresses the problem of inaccurate solutions in scientific computing for researchers and engineers, representing a strong specific gain rather than a broad paradigm shift.
The paper tackles the challenge of low accuracy in physics-informed neural networks (PINNs) by proposing energy natural gradient descent, which reduces errors by several orders of magnitude compared to standard optimizers like gradient descent or Adam.
We propose energy natural gradient descent, a natural gradient method with respect to a Hessian-induced Riemannian metric as an optimization algorithm for physics-informed neural networks (PINNs) and the deep Ritz method. As a main motivation we show that the update direction in function space resulting from the energy natural gradient corresponds to the Newton direction modulo an orthogonal projection onto the model's tangent space. We demonstrate experimentally that energy natural gradient descent yields highly accurate solutions with errors several orders of magnitude smaller than what is obtained when training PINNs with standard optimizers like gradient descent or Adam, even when those are allowed significantly more computation time.