Pacmann AI Project

sampling: Non-adaptive and adaptive sampling based on residuals for Neural networks based on physical information

@github:zulkarnainprastyo, @medium:Zulkarnain Prastyo

Framework to Describe These Projects

1. Goal: The primary objective of the project was to explore and apply various sampling methods in Physically Informed Neural Networks (PINN) for solving partial differential equations (PDEs). The project aimed to enhance the accuracy and efficiency of PINN solutions by developing novel adaptive sampling strategies.

2. Impact: Through the study of two types of PINN, this project made significant strides in improving the accuracy of PDE solutions. The introduction of new residual-based adaptive sampling algorithms, RAD and RAR-D, led to a substantial increase in accuracy, particularly in cases with low residual points. These strategies showed promise in addressing complex PDEs with intricate solutions.

3. Challenges: The project encountered challenges in designing and implementing the adaptive sampling algorithms. Balancing the location points with large and small PDE residuals while optimizing hyperparameters was a key challenge. Additionally, the selection of suitable sampling methods for high-dimensional problems posed difficulties.

4. Interesting Findings: The project yielded several interesting findings: • RAD with specific hyperparameter values (k = 1 and c = 1) emerged as a robust default sampling method for solving new PDEs. Fine-tuning k and c allowed for effective balancing of location points. • RAR-D, with default values (k = 2 and c = 0), proved to be a computationally efficient alternative to RAD while maintaining comparable accuracy. • Random-R was identified as a useful approach in cases where adaptive sampling was not feasible due to constraints in extracting residual points using probability density functions. • The choice of sampling sequences, such as Hammer three, was recommended for obtaining fixed points in cases where a low-difference sequence was needed.

Conclusion: The project underscored the significance of adaptive sampling in enhancing the accuracy of PINN solutions for complex PDEs. The proposed RAD and RAR-D algorithms, along with the insights gained, contribute to the advancement of Physically Informed Neural Networks in solving real-world problems. While the project utilized a brute-force approach for sampling, future endeavors could explore advanced techniques like Generative Adversarial Networks (GANs) for high-dimensional scenarios.

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