Understanding and mitigating gradient pathologies in physics-informed neural networks (2001.04536v1)

Abstract: The widespread use of neural networks across different scientific domains often involves constraining them to satisfy certain symmetries, conservation laws, or other domain knowledge. Such constraints are often imposed as soft penalties during model training and effectively act as domain-specific regularizers of the empirical risk loss. Physics-informed neural networks is an example of this philosophy in which the outputs of deep neural networks are constrained to approximately satisfy a given set of partial differential equations. In this work we review recent advances in scientific machine learning with a specific focus on the effectiveness of physics-informed neural networks in predicting outcomes of physical systems and discovering hidden physics from noisy data. We will also identify and analyze a fundamental mode of failure of such approaches that is related to numerical stiffness leading to unbalanced back-propagated gradients during model training. To address this limitation we present a learning rate annealing algorithm that utilizes gradient statistics during model training to balance the interplay between different terms in composite loss functions. We also propose a novel neural network architecture that is more resilient to such gradient pathologies. Taken together, our developments provide new insights into the training of constrained neural networks and consistently improve the predictive accuracy of physics-informed neural networks by a factor of 50-100x across a range of problems in computational physics. All code and data accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/GradientPathologiesPINNs}.

Understanding and Mitigating Gradient Pathologies in Physics-Informed Neural Networks

The paper investigates gradient pathologies in Physics-Informed Neural Networks (PINNs) and proposes methods to address these issues that frequently arise during model training. The focus is on understanding the intrinsic difficulties associated with learning partial differential equations (PDEs) using neural networks, particularly in the context of computational physics, where PINNs are regularly employed.

Key Contributions

1. Identification of Gradient Pathologies: The authors identify major failure modes in PINNs associated with stiffness in the gradient flow dynamics. These pathologies are responsible for unbalanced gradients during training, leading to inaccuracies in model predictions. Specifically, such issues are evident when using conventional gradient descent techniques, which often struggle with the unique characteristics of PINN loss functions composed of PDE constraints and data-fitting terms.
2. Learning Rate Annealing Algorithm: To rectify the imbalance in gradient magnitudes, the paper introduces an adaptive learning rate annealing algorithm. By adjusting learning rates in response to the gradient magnitudes of individual loss terms, this method effectively balances data-fitting against equation constraints, leading to improved model training and predictive accuracy.
3. Improved Neural Network Architecture: An alternative neural network architecture is proposed, designed to reduce stiffness in the gradient flow dynamics, thus aiding in the stability and accuracy of PINN models. This model incorporates mechanisms similar to neural attention, enhancing its ability to capture complex patterns inherent in physics-based problems.
4. Empirical Evaluation: The paper presents a thorough empirical evaluation across several computational physics benchmarks. The proposed algorithm and architecture yield improvements by a factor of 50-100x in predictive accuracy compared to traditional PINNs. This improvement is consistent across various test cases, including the Helmholtz equation, Klein-Gordon equation, and flow in a lid-driven cavity.

Implications and Future Directions

The findings have significant implications for the field of scientific machine learning, particularly in the field of solving PDEs with neural networks. The paper presents compelling evidence that the complexities of PINNs necessitate tailored optimization strategies and architectural innovations.

• Practical Implications: By improving the training efficiency and accuracy of PINNs, the proposed methods may broaden the applicability of neural networks in engineering and physics, enabling the solution of more complex systems where data may be sparse or noisy.
• Theoretical Implications: The work prompts further investigation into the theoretical underpinnings of gradient flow dynamics in PINNs. Analyzing the connections between the stiffness of PDEs and the training dynamics could yield new theoretical insights, potentially resulting in alternative training algorithms more suited to these problems.
• Future Research Directions: There are several open avenues for extending this work. Future research could explore more stable discretizations for gradient flow dynamics, alternative neural architectures tailored specifically for physical systems, and extending PINNs' utility in multi-task learning contexts. Such efforts would require multidisciplinary collaboration, leveraging areas like deep learning optimization, dynamical systems, and numerical analysis.

In summary, the paper presents a detailed exploration of the challenges associated with PINN training and offers concrete methods to enhance model performance. These contributions pave the way for more robust applications of neural networks in scientific domains, pushing the boundaries of what can be computationally achieved through machine learning in computational physics.