Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers (2007.00016v2)

Abstract: Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting for effects not captured by the discretized PDE. We target the problem of reducing numerical errors of iterative PDE solvers and compare different learning approaches for finding complex correction functions. We find that previously used learning approaches are significantly outperformed by methods that integrate the solver into the training loop and thereby allow the model to interact with the PDE during training. This provides the model with realistic input distributions that take previous corrections into account, yielding improvements in accuracy with stable rollouts of several hundred recurrent evaluation steps and surpassing even tailored supervised variants. We highlight the performance of the differentiable physics networks for a wide variety of PDEs, from non-linear advection-diffusion systems to three-dimensional Navier-Stokes flows.

• The paper introduces a solver-in-the-loop framework that integrates machine learning with PDE solvers to dynamically correct numerical errors.
• It employs differentiable physics during training across various PDEs, including non-linear advection-diffusion and 3D Navier-Stokes flows, to enhance stability.
• Empirical evaluations demonstrate significant error reductions over traditional methods, promising transformative improvements in high-fidelity simulations.

Analyzing Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers

The paper "Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers" investigates a significant area in computational science: enhancing the accuracy of partial differential equation (PDE) solvers through machine learning techniques. In many scientific and engineering domains, obtaining precise solutions to PDEs is essential, yet computationally expensive. This paper proposes an innovative approach by leveraging machine learning models integrated with iterative PDE solvers, aiming to correct the numerical errors during the discretization of PDEs.

Key Contributions

1. Differentiable Physics for Training: The research emphasizes integrating a PDE solver within the training loop of machine learning models, thereby allowing these models to dynamically interact with the PDEs during training. This approach benefits from realistic input distributions that reflect previous corrections, enhancing the model's accuracy and stability over multiple recurrent evaluation steps. The paper demonstrates the superior performance of this method over traditional supervised learning approaches that rely on pre-computed data, which often fails to generalize well due to distribution shifts in the error accumulation process.
2. Variety of PDEs: The work covers a broad array of PDEs including non-linear advection-diffusion systems and three-dimensional Navier-Stokes flows, highlighting the versatility and robustness of the proposed learning framework across different numerical challenges.
3. Empirical Evaluations: Detailed empirical evaluations are presented, showcasing the accuracy improvements achieved by employing differentiable physics in training scenarios. The research reports significant reductions in numerical errors beyond those achieved by baseline models, such as the supervised learning techniques and pre-computed correction methods.

Technical and Practical Implications

The paper presents a methodological advancement with the potential to substantially augment the numerical accuracy offered by traditional solvers. By embedding machine learning models within the solver loop, the approach mitigates the growing discrepancies between source and reference solutions, which commonly propagate under distribution shifts in supervised models. This direct interaction mechanism allows the model to adapt to variable physics closely, extending generalization capabilities and yielding enhanced solution accuracy.

For computational fluid dynamics (CFD) and other PDE-based simulations, this hybrid solver approach can be transformative. Real-world applications, especially those requiring rapid yet precise simulations, such as aerodynamics or climate modeling, could benefit from these improvements in error correction. Additionally, this paper underscores the potential of combining ANNs with iterative solvers, possibly setting the stage for a shift towards hybrid computational methods in engineering simulations.

Future Directions

Overall, the exploration pioneered by this research sets a precedent for future work, suggesting several avenues for further paper:

• Efficiency and Scalability: Research could explore optimizing such frameworks for greater computational efficiency and scalability to larger, more complex systems.
• Integration with Diverse Solvers: Extending this methodology to encompass more diverse types of solvers and PDEs, including those outside the demonstrated advection-diffusion and Navier-Stokes systems, might broaden applicability.
• Advanced Learning Techniques: Incorporating advanced learning paradigms like reinforcement learning or variational methods might yield even higher accuracy and adaptability to unseen physical scenarios.
• Hardware Optimization: As hardware evolves, so too should the integration and optimization of neural networks, especially on new architectures conducive to machine learning inference.

Ultimately, this research contributes a novel approach to overcoming a longstanding challenge in numerical solution accuracy. While impressive improvements in accuracy are documented, particularly through solver-in-the-loop configurations, practical deployment in various industrial domains will serve as a strong validation for such hybrid methods. The work proposes a promising future where learned corrections enhance the traditionally deterministic field of PDE solvers, creating more efficient, adaptable, and accurate computational systems.