# A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecture

arXiv: https://arxiv.org/abs/2005.00615 · HTML: https://mlsanity.com/paper/2005.00615/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 May 2020.

## Built on by (most cited first, 30 of 50)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
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- [ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2408.11104/) (2024)
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- [Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorial](https://mlsanity.com/paper/2305.04933/) (2023)
- [Adaptive Self-supervision Algorithms for Physics-informed Neural Networks](https://mlsanity.com/paper/2207.04084/) (2022)
- [Physics-Informed Neural Networks with Adaptive Localized Artificial Viscosity](https://mlsanity.com/paper/2203.08802/) (2022)
- [PIXEL: Physics-Informed Cell Representations for Fast and Accurate PDE Solvers](https://mlsanity.com/paper/2207.12800/) (2022)
- [Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks](https://mlsanity.com/paper/2407.01613/) (2024)
- [Improved Training of Physics-Informed Neural Networks with Model Ensembles](https://mlsanity.com/paper/2204.05108/) (2022)
- [Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs)](https://mlsanity.com/paper/2209.09988/) (2022)
- [Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantification](https://mlsanity.com/paper/2402.15115/) (2024)
- [An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks](https://mlsanity.com/paper/2306.04904/) (2023)
- [A practical PINN framework for multi-scale problems with multi-magnitude loss terms](https://mlsanity.com/paper/2308.06672/) (2023)
- [RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networks](https://mlsanity.com/paper/2205.01051/) (2022)
- [Solving High Frequency and Multi-Scale PDEs with Gaussian Processes](https://mlsanity.com/paper/2311.04465/) (2023)
- [Learning in PINNs: Phase transition, diffusion equilibrium, and generalization](https://mlsanity.com/paper/2403.18494/) (2024)
- [Spectral operator learning for parametric PDEs without data reliance](https://mlsanity.com/paper/2310.02013/) (2023)
- [Parametric Encoding with Attention and Convolution Mitigate Spectral Bias of Neural Partial Differential Equation Solvers](https://mlsanity.com/paper/2403.15652/) (2024)
- [Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equations](https://mlsanity.com/paper/2303.11577/) (2023)
- [Towards Complex Dynamic Physics System Simulation with Graph Neural ODEs](https://mlsanity.com/paper/2305.12334/) (2023)
- [Fourier PINNs: From Strong Boundary Conditions to Adaptive Fourier Bases](https://mlsanity.com/paper/2410.03496/) (2024)
- [Self-adaptive weighting and sampling for physics-informed neural networks](https://mlsanity.com/paper/2511.05452/) (2025)
- [Deep learning for full-field ultrasonic characterization](https://mlsanity.com/paper/2301.02378/) (2023)
- [Dual-Balancing for Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.11117/) (2025)
- [Robust Regression with Highly Corrupted Data via Physics Informed Neural Networks](https://mlsanity.com/paper/2210.10646/) (2022)
- [Splitting physics-informed neural networks for inferring the dynamics of integer- and fractional-order neuron models](https://mlsanity.com/paper/2304.13205/) (2023)
- [Temporal Consistency Loss for Physics-Informed Neural Networks](https://mlsanity.com/paper/2301.13262/) (2023)

## Abstract (arXiv)

Data sparsity is a common issue to train machine learning tools such as neural networks for engineering and scientific applications, where experiments and simulations are expensive. Recently physics-constrained neural networks (PCNNs) were developed to reduce the required amount of training data. However, the weights of different losses from data and physical constraints are adjusted empirically in PCNNs. In this paper, a new physics-constrained neural network with the minimax architecture (PCNN-MM) is proposed so that the weights of different losses can be adjusted systematically. The training of the PCNN-MM is searching the high-order saddle points of the objective function. A novel saddle point search algorithm called Dual-Dimer method is developed. It is demonstrated that the Dual-Dimer method is computationally more efficient than the gradient descent ascent method for nonconvex-nonconcave functions and provides additional eigenvalue information to verify search results. A heat transfer example also shows that the convergence of PCNN-MMs is faster than that of traditional PCNNs.