Multi-stage Neural Networks: Function Approximator of Machine Precision
This addresses the need for high-accuracy function approximation in scientific computing, representing a significant improvement over standard neural networks.
The paper tackles the problem of neural networks struggling to achieve high precision in scientific applications, introducing multi-stage neural networks that reduce prediction errors to nearly machine-precision levels, such as O(10^{-16}) for double-floating point, in regression and physics-informed tasks.
Deep learning techniques are increasingly applied to scientific problems, where the precision of networks is crucial. Despite being deemed as universal function approximators, neural networks, in practice, struggle to reduce the prediction errors below $O(10^{-5})$ even with large network size and extended training iterations. To address this issue, we developed the multi-stage neural networks that divides the training process into different stages, with each stage using a new network that is optimized to fit the residue from the previous stage. Across successive stages, the residue magnitudes decreases substantially and follows an inverse power-law relationship with the residue frequencies. The multi-stage neural networks effectively mitigate the spectral biases associated with regular neural networks, enabling them to capture the high frequency feature of target functions. We demonstrate that the prediction error from the multi-stage training for both regression problems and physics-informed neural networks can nearly reach the machine-precision $O(10^{-16})$ of double-floating point within a finite number of iterations. Such levels of accuracy are rarely attainable using single neural networks alone.