# Accelerating Physics-Informed Neural Network Training with Prior Dictionaries

arXiv: https://arxiv.org/abs/2004.08151 · HTML: https://mlsanity.com/paper/2004.08151/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Apr 2020.

## Built on by (most cited first, 17 of 17)

- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Overview frequency principle/spectral bias in deep learning](https://mlsanity.com/paper/2201.07395/) (2022)
- [A novel meta-learning initialization method for physics-informed neural networks](https://mlsanity.com/paper/2107.10991/) (2021)
- [Data vs. Physics: The Apparent Pareto Front of Physics-Informed Neural Networks](https://mlsanity.com/paper/2105.00862/) (2021)
- [MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networks](https://mlsanity.com/paper/2306.02816/) (2023)
- [An extended physics informed neural network for preliminary analysis of parametric optimal control problems](https://mlsanity.com/paper/2110.13530/) (2021)
- [Physical Activation Functions (PAFs): An Approach for More Efficient Induction of Physics into Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2205.14630/) (2022)
- [Adaptive Physics-informed Neural Networks: A Survey](https://mlsanity.com/paper/2503.18181/) (2025)
- [An Upper Limit of Decaying Rate with Respect to Frequency in Deep Neural Network](https://mlsanity.com/paper/2105.11675/) (2021)
- [A Generalizable Physics-informed Learning Framework for Risk Probability Estimation](https://mlsanity.com/paper/2305.06432/) (2023)
- [Up-sampling-only and Adaptive Mesh-based GNN for Simulating Physical Systems](https://mlsanity.com/paper/2409.04740/) (2024)
- [AMS-Net: Adaptive Multiscale Sparse Neural Network with Interpretable Basis Expansion for Multiphase Flow Problems](https://mlsanity.com/paper/2207.11735/) (2022)
- [Generalizable Physics-Informed Learning for Stochastic Safety-Critical Systems](https://mlsanity.com/paper/2407.08868/) (2024)
- [VI-PINNs: Variance-involved Physics-informed Neural Networks for Fast and Accurate Prediction of Partial Differential Equations](https://mlsanity.com/paper/2211.16753/) (2022)
- [Physics-Informed Deep B-Spline Networks](https://mlsanity.com/paper/2503.16777/) (2025)
- [Transport-Embedded Neural Architecture: Redefining the Landscape of physics aware neural models in fluid mechanics](https://mlsanity.com/paper/2410.04114/) (2024)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) can be regarded as general-purpose PDE solvers, but it might be slow to train PINNs on particular problems, and there is no theoretical guarantee of corresponding error bounds. In this manuscript, we propose a variant called Prior Dictionary based Physics-Informed Neural Networks (PD-PINNs). Equipped with task-dependent dictionaries, PD-PINNs enjoy enhanced representation power on the tasks, which helps to capture features provided by dictionaries so that the proposed neural networks can achieve faster convergence in the process of training. In various numerical simulations, compared with existing PINN methods, combining prior dictionaries can significantly enhance convergence speed. In terms of theory, we obtain the error bounds applicable to PINNs and PD-PINNs for solving elliptic partial differential equations of second order. It is proved that under certain mild conditions, the prediction error made by neural networks can be bounded by expected loss of PDEs and boundary conditions.