Efficient Training of Physics-Informed Neural Networks through Joint Optimization of Collocation and Experimental Data Points

Kernekoncepter

This work introduces PINNACLE, the first algorithm that jointly optimizes the selection of all training point types for Physics-Informed Neural Networks (PINNs), including collocation points for enforcing PDEs and initial/boundary conditions, as well as experimental data points. PINNACLE automatically adjusts the proportion of collocation point types as training progresses to boost PINN performance.

Resumé

The paper introduces PINNACLE, a novel algorithm for efficiently training Physics-Informed Neural Networks (PINNs). PINNs incorporate partial differential equations (PDEs) and their initial/boundary conditions as soft constraints during training, requiring multiple types of training points: collocation points to enforce the PDE and boundary conditions, and experimental points providing ground truth solution values.
The key insights are:

The authors define an augmented input space that jointly represents all training point types, enabling analysis of their cross-interactions during PINN training.
Using the Neural Tangent Kernel (NTK) eigenspectrum of this augmented space, the authors define a "convergence degree" criterion that captures how well a set of training points would improve PINN training convergence.
The PINNACLE algorithm then selects training points that maximize this convergence degree, automatically adjusting the proportion of collocation point types as training progresses.

The authors show that PINNACLE outperforms existing point selection methods for forward, inverse, and transfer learning problems involving various PDEs. The selected training points are also interpretable, resembling heuristics used in past works. Overall, PINNACLE is the first method to jointly optimize all training point types for PINNs, leading to significant performance improvements.

Statistik

The paper does not contain any explicit numerical data or statistics. The key results are presented through plots of prediction errors and parameter estimates.

Citater

"PINNACLE uses information on the interaction among training point types, which had not been considered before, based on an analysis of PINN training dynamics via the Neural Tangent Kernel (NTK)."
"We theoretically show that the criterion used by PINNACLE is related to the PINN generalization error, and empirically demonstrate that PINNACLE is able to outperform existing point selection methods for forward, inverse, and transfer learning problems."

Vigtigste indsigter udtrukket fra

PINNACLE

by Gregory Kang... kl. arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07662.pdf

Dybere Forespørgsler

How can the PINNACLE framework be extended to handle other types of constraints beyond PDEs, such as conservation laws or other physical principles

The PINNACLE framework can be extended to handle other types of constraints beyond PDEs by incorporating additional physics principles or constraints into the composite loss function. For example, conservation laws can be included as soft constraints in the loss function, similar to how PDEs are currently integrated. By defining the augmented input space to include variables related to these additional constraints, the PINNACLE algorithm can jointly optimize the selection of training points to satisfy all constraints simultaneously. This would involve expanding the analysis of training dynamics using the NTK to incorporate the interactions among the various types of constraints, allowing for a more comprehensive and efficient training process.

Can the joint optimization of training points be combined with other techniques like adaptive activation functions or loss weighting to further improve PINN training

The joint optimization of training points in the PINNACLE framework can indeed be combined with other techniques like adaptive activation functions or loss weighting to further improve PINN training. By integrating adaptive activation functions, the network can dynamically adjust its activation functions based on the selected training points, enhancing the network's ability to capture complex patterns and relationships in the data. Additionally, incorporating loss weighting techniques can help prioritize certain types of training points or constraints during the training process, leading to more effective learning and convergence. By combining these techniques with the joint optimization of training points, the PINNACLE framework can achieve even better performance and generalization in PINNs.

What are the potential applications of the augmented input space representation and NTK-based analysis beyond the PINNACLE algorithm, such as in other physics-informed or multi-task learning settings

The augmented input space representation and NTK-based analysis used in the PINNACLE algorithm have broad applications beyond just PINNs. In other physics-informed learning settings, such as solving inverse problems or learning governing equations from data, the augmented input space can help in jointly optimizing the selection of training points to improve model performance. Additionally, the NTK-based analysis can provide insights into the training dynamics of neural networks in various multi-task learning scenarios. By analyzing the interactions among different types of tasks or constraints, the NTK can guide the selection of training points and the design of the network architecture to enhance learning efficiency and generalization across multiple tasks. This approach can be valuable in a wide range of scientific and engineering applications where domain knowledge and constraints play a crucial role in model training and inference.