Exactly conservative physics-informed neural networks and deep operator networks for dynamical systems

Area of Science:

• Computational mathematics
• Applied mathematics
• Scientific machine learning

Background:

• Dynamical systems and ordinary differential equations (ODEs) are fundamental in science.
• Physics-informed neural networks (PINNs) offer a powerful tool for solving ODEs.
• Ensuring conservation laws in neural network solvers is crucial for accuracy.

Purpose of the Study:

• To introduce a novel method for training exactly conservative PINNs and physics-informed deep operator networks (IPDONs).
• To enhance the accuracy and stability of neural network solvers for dynamical systems.
• To demonstrate the superiority of conservative solvers over non-conservative ones.

Main Methods:

• A projection-based technique is employed to map candidate solutions onto invariant manifolds.
• The method ensures that the neural network solutions strictly adhere to the system's first integrals.
• The approach is applicable to dynamical systems possessing at least one first integral.

Main Results:

• Exactly conservative PINNs and IPDONs demonstrate vastly superior performance compared to non-conservative methods.
• The projection technique effectively enforces conservation laws during the training process.
• Significant improvements in accuracy and stability were observed across several real-world mathematical problems.

Conclusions:

• The proposed projection-based method provides a robust framework for developing highly accurate conservative neural network solvers.
• This advancement is critical for reliable simulations of dynamical systems in various scientific domains.
• The findings highlight the importance of incorporating physical conservation laws into neural network architectures for scientific computing.