ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks (2408.11104v3)

Abstract: The loss functions of many learning problems contain multiple additive terms that can disagree and yield conflicting update directions. For Physics-Informed Neural Networks (PINNs), loss terms on initial/boundary conditions and physics equations are particularly interesting as they are well-established as highly difficult tasks. To improve learning the challenging multi-objective task posed by PINNs, we propose the ConFIG method, which provides conflict-free updates by ensuring a positive dot product between the final update and each loss-specific gradient. It also maintains consistent optimization rates for all loss terms and dynamically adjusts gradient magnitudes based on conflict levels. We additionally leverage momentum to accelerate optimizations by alternating the back-propagation of different loss terms. We provide a mathematical proof showing the convergence of the ConFIG method, and it is evaluated across a range of challenging PINN scenarios. ConFIG consistently shows superior performance and runtime compared to baseline methods. We also test the proposed method in a classic multi-task benchmark, where the ConFIG method likewise exhibits a highly promising performance. Source code is available at https://tum-pbs.github.io/ConFIG

• The paper introduces Conflict-Free Inverse Gradients (ConFIG) to resolve conflicting gradient directions in Physics-Informed Neural Networks.
• It employs dynamic scaling and adaptive gradient magnitudes to ensure balanced optimization across multiple loss terms.
• Experiments on various PDEs and multi-task learning benchmarks demonstrate improved convergence speed and computational efficiency over traditional methods.

ConFIG: Towards Conflict-Free Training of Physics-Informed Neural Networks

The paper "ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks" introduces a novel approach called Conflict-Free Inverse Gradients (ConFIG) to handle the training challenges of Physics-Informed Neural Networks (PINNs). The authors address issues arising from conflicting gradient directions during the optimization process, which is particularly problematic in PINNs due to the presence of multiple loss terms derived from initial/boundary conditions and physics equations.

Key Contributions and Methodology

Physics-Informed Neural Networks (PINNs) have gained traction for their ability to solve partial differential equations (PDEs) by embedding physical laws directly into the neural network's loss function. Despite their promise, training PINNs is notoriously difficult due to disparate update directions from different loss terms. Traditional approaches often involve heuristic weighting strategies, but they lack a consensus on optimal methods.

The ConFIG methodology presents a structured approach to obtain conflict-free updates, ensuring the final gradient aligns positively with each individual loss-specific gradient. This is achieved through inverse operations to standardize the projection length of the final gradient on each loss term, dynamically scaling based on the degree of gradient conflict. The ConFIG approach can be summarized in terms of the following properties:

1. Conflict-Free Update Directions: The update gradient does not conflict with any loss-specific gradients.
2. Uniform Optimization Rates: The projection length on each loss-specific gradient is uniform, promoting balanced optimization.
3. Adaptive Gradient Magnitude: The magnitude of the final gradient is scaled dynamically, enhancing convergence irrespective of conflict intensity.

Additionally, the authors introduce an enhanced variant, M-ConFIG, leveraging momentum to accelerate optimizations by alternating between different loss-specific gradients. This variant dramatically reduces computational costs while maintaining effectiveness.

Experiments and Results

Physics-Informed Neural Networks (PINNs)

The paper evaluates ConFIG and M-ConFIG across various PDE scenarios, including the 1D Burgers equations, 1D Schrödinger equation, 2D Kovasznay flow, and 3D Beltrami flow, showing notable improvements compared to baseline and state-of-the-art methods.

• Two Loss Terms: When training PINNs with two loss terms (e.g., combining boundary and initial conditions), ConFIG and PCGrad consistently outperform baselines such as Adam and other heuristic methods. The ConFIG method demonstrates superior performance owing to its ability to harmonize gradient contributions effectively.
• Three Loss Terms: In scenarios involving three loss terms (e.g., boundary, initial, and PDE residuals), ConFIG again excels. Although PCGrad shows competitive performance, ConFIG's dynamic scaling of gradient magnitudes enables more robust convergence, particularly evident in complex domains like the Beltrami flow.

Multi-Task Learning (MTL)

Expanding the application domain, the paper also explores ConFIG in a standard Mult-Task Learning benchmark using the CelebA dataset. The results substantiate the general applicability of ConFIG beyond PINNs, demonstrating marked improvements in mean rank (MR) and average F1 score ($\bar{F_1}$).

• Scalability: The M-ConFIG variant showcases enhanced efficiency, lowering computational overhead while retaining efficacy. This makes ConFIG highly suitable for large-scale MTL tasks.

Implications and Future Work

The ConFIG method addresses a significant bottleneck in training PINNs by mitigating gradient conflicts, which are a primary source of inefficiency and suboptimal convergence. The demonstrated improvements in both runtime efficiency and accuracy across diverse tasks underscore the potential of this approach to redefine training paradigms for PINNs and multi-task learning frameworks.

Looking forward, further refinement of the M-ConFIG method, particularly in handling a larger number of loss terms without performance degradation, remains a promising avenue for research. Additionally, exploring ConFIG's efficacy in more complex, real-world applications could provide deeper insights and broader applicability.

In conclusion, the ConFIG methodology represents a significant advancement in the optimization of neural networks dealing with multi-objective learning problems. Its ability to harmonize conflicting gradients and adaptively scale optimization steps holds substantial promise for future developments in AI and machine learning. The practical and theoretical implications highlighted by this research pave the way for more robust and efficient learning models across varied domains.