Advancing Generalization in PINNs Through Latent-Space Representations.

Area of Science:

• Computational science
• Applied mathematics
• Machine learning

Background:

• Physics-informed neural networks (PINNs) are effective for modeling dynamical systems governed by partial differential equations (PDEs).
• However, existing PINNs exhibit limited generalization capabilities across different scenarios, such as varying initial conditions or PDE coefficients.

Purpose of the Study:

• To introduce a novel physics-informed neural PDE solver, the physics-informed dynamics representation learner (PiDo), designed for enhanced generalization across diverse PDE configurations.
• To address challenges in integrating latent dynamics models within physics-informed frameworks, improving optimization and stability.

Main Methods:

• PiDo projects PDE solutions into a latent space using auto-decoding to exploit shared dynamical system structures.
• It learns latent representation dynamics conditioned on PDE coefficients.
• Novel regularization techniques are employed to diagnose and mitigate optimization difficulties within the latent space.

Main Results:

• PiDo demonstrates effective generalization across varying initial conditions, PDE coefficients, and training-time horizons.
• The approach shows enhanced temporal extrapolation performance and improved training stability.
• Validated on 1-D combined equations and 2-D Navier-Stokes equations.

Conclusions:

• PiDo offers a robust framework for physics-informed PDE solving with superior generalization capabilities.
• Learned representations are transferable to downstream tasks like long-term integration and inverse problems.
• The developed regularization strategy effectively addresses optimization challenges in latent-space physics-informed learning.