DOF: Accelerating High-order Differential Operators with Forward Propagation (2402.09730v1)

Abstract: Solving partial differential equations (PDEs) efficiently is essential for analyzing complex physical systems. Recent advancements in leveraging deep learning for solving PDE have shown significant promise. However, machine learning methods, such as Physics-Informed Neural Networks (PINN), face challenges in handling high-order derivatives of neural network-parameterized functions. Inspired by Forward Laplacian, a recent method of accelerating Laplacian computation, we propose an efficient computational framework, Differential Operator with Forward-propagation (DOF), for calculating general second-order differential operators without losing any precision. We provide rigorous proof of the advantages of our method over existing methods, demonstrating two times improvement in efficiency and reduced memory consumption on any architectures. Empirical results illustrate that our method surpasses traditional automatic differentiation (AutoDiff) techniques, achieving 2x improvement on the MLP structure and nearly 20x improvement on the MLP with Jacobian sparsity.

• The paper introduces the DOF method that leverages forward propagation to compute general second-order differential operators more efficiently than Hessian-based methods.
• Empirical tests demonstrate up to 2x faster computation times and nearly 20x lower memory usage across various neural network architectures.
• The framework broadens the capabilities of neural PDE solvers and lays the groundwork for future innovations in efficient derivative computation.

Accelerating High-Order Differential Operators with Forward Propagation for Neural Network-Based PDE Solvers

Introduction

Partial differential equations (PDEs) are fundamental in modeling natural phenomena and engineering challenges. Neural networks (NNs), particularly physics-informed neural networks (PINNs), have gained traction in solving PDEs due to their ability to parameterize solutions to complex problems directly. However, efficiently computing high-order derivatives, key to solving numerous PDEs, remains cumbersome with current automatic differentiation (AutoDiff) methods due to significant computational and memory overheads. This paper introduces the Differential Operator with Forward-propagation (DOF) framework, a novel approach for calculating general second-order differential operators efficiently, extending the capabilities of neural network-based solvers for a broader class of PDEs.

Methodology

The introduced DOF method capitalizes on the approach taken by Forward Laplacian (FL) but extends its applicability to general second-order differential operators without sacrificing precision. DOF operates under a similar computational paradigm as FL, employing efficient forward propagation techniques to compute the desired operators.

The framework specifics involve decomposing the coefficient matrix of the differential operator and leveraging forward propagation to compute necessary derivatives, resulting in notable computational and memory efficiencies across various neural network architectures. It significantly outperforms conventional AutoDiff techniques in both theoretical and empirical evaluations. Two theorems central to the framework assert that DOF can halve the computational cost and reduce memory consumption compared to Hessian-based methods, irrespective of the neural network's architecture.

Results

Empirical tests confirm the theoretical promises of DOF. Experiments using multi-layer perceptrons (MLPs) and MLPs with Jacobian sparsity—representing different NN architectures—showcase DOF's superior performance. Specifically, improvements include up to 2x faster computation times and nearly 20x memory usage reductions on specialized architectures. These benchmarks span various operators (e.g., elliptic, low-rank), illustrating DOF's broad applicability and efficiency gains across different scenarios.

Future Implications

Practically, DOF heralds a significant step forward in neural network-based PDE solvers, with immediate applications in physics, engineering, and beyond. Theoretically, it opens new avenues for exploring efficient derivative computation methods in deep learning, potentially inspiring further innovations in NN architecture designs and optimization algorithms. Looking ahead, adapting DOF to higher-order differential operators or integrating it with emerging NN-based PDE solver frameworks could further expand its utility, paving the way for tackling an even wider array of complex PDEs.

Conclusion

In summary, the Differential Operator with Forward-propagation framework sets a new standard for computing second-order differential operators within neural network-based PDE solvers. By efficiently leveraging forward propagation, DOF not only reduces computational and memory demands but also expands the potential of NN-based methods in solving intricate differential equations. Future research inspired by DOF could lead to more sophisticated solvers capable of addressing a broader spectrum of scientific and engineering challenges, marking a notable advance in the merging worlds of artificial intelligence and numerical analysis.