ODEN: A Framework to Solve Ordinary Differential Equations using Artificial Neural Networks
This provides a novel method for computational scientists and engineers to solve ODEs, though it is incremental as it builds on existing neural network approaches.
The paper tackles solving ordinary differential equations using feedforward neural networks, showing they can outperform traditional numerical techniques by proving a loss function that doesn't require exact solutions and identifying optimal architectures matching equation complexity.
We explore in detail a method to solve ordinary differential equations using feedforward neural networks. We prove a specific loss function, which does not require knowledge of the exact solution, to be a suitable standard metric to evaluate neural networks' performance. Neural networks are shown to be proficient at approximating continuous solutions within their training domains. We illustrate neural networks' ability to outperform traditional standard numerical techniques. Training is thoroughly examined and three universal phases are found: (i) a prior tangent adjustment, (ii) a curvature fitting, and (iii) a fine-tuning stage. The main limitation of the method is the nontrivial task of finding the appropriate neural network architecture and the choice of neural network hyperparameters for efficient optimization. However, we observe an optimal architecture that matches the complexity of the differential equation. A user-friendly and adaptable open-source code (ODE$\mathcal{N}$) is provided on GitHub.