Experimental Demonstration of an Optical Neural PDE Solver via On-Chip PINN Training (2501.00742v1)

Abstract: Partial differential equation (PDE) is an important math tool in science and engineering. This paper experimentally demonstrates an optical neural PDE solver by leveraging the back-propagation-free on-photonic-chip training of physics-informed neural networks.

• The paper introduces a novel on-chip PINN training approach that eliminates back propagation, reducing computational overhead in solving PDEs.
• The methodology employs zeroth-order optimization on a micro-ring resonator weight bank to achieve a low ℓ2 error in a 1D heat equation model.
• Experimental results validate the approach, paving the way for energy-efficient, real-time PDE solving on photonic hardware for edge computing.

Optical Neural PDE Solver with On-Chip PINN Training

The paper presents a novel approach to solving partial differential equations (PDEs) using an optical neural network (ONN) implemented with back-propagation-free training on a photonics chip. This research explores the potential of physics-informed neural networks (PINNs) trained on hardware well-suited for high-throughput and energy-efficient computation. The authors argue that on-chip PINN training addresses the limitations of traditional PDE solvers and computationally intensive PINN implementations on digital devices.

Background and Motivation

The paper acknowledges the pivotal role PDEs play in a variety of scientific and engineering disciplines, such as electromagnetic modeling, thermal analysis, and medical imaging. Traditional discretization-based solvers, like finite-difference and finite-element methods, are resource-intensive and require extensive computation time. PINNs provide an alternative by approximating PDE solutions without the need for discretization, yet training them typically demands substantial computational power. The integration of PINN training on photonics hardware aims to exploit the advantages of ONNs, namely reduced energy consumption and increased parallelism, to enable real-time PDE solutions on edge devices.

Methodology

The paper employs a back-propagation-free training method tailored for photonics chips to sidestep challenges related to implementing back propagation (BP) in optical systems—such as extra memory requirements and dealing with non-differentiable elements due to hardware imperfections. By adopting a zeroth-order (ZO) optimization approach, the authors minimize the training loss of a PINN by iteratively adjusting the tunable parameters of photonic devices. This method mitigates the need for a pre-calibration process, allowing for real-time adjustments to fabrication errors and environmental noise on the chip.

The experimental setup features a 1×4 micro-ring resonator (MRR) weight bank, which facilitates on-chip training of a neural network to learn the solution to a one-dimensional heat equation. The training demonstrates a rapid learning curve, achieving a remarkably low $\ell_2$ error, verifying the feasibility of deploying such systems for solving simple PDEs.

Experimental Results and Implications

The efficacy of the proposed method is validated through both simulation and physical hardware demonstrations. Simulation results illuminate the impact that hardware precision has on training outcome, showing that reduced bit accuracy hinders the network’s ability to learn governing physics accurately. Moreover, the experimental results pinpoint successful training on an actual photonic chip, with a performance consistent with a theoretical bit accuracy between 8-bits and 10-bits.

The implications of this research are multifaceted, offering a new paradigm for integrating deep learning-based PDE solvers with photonic computing. Scaling this approach further could significantly benefit real-time processing tasks in applications ranging from environmental modeling to signal processing.

Future Directions

To extend the reach of this work, future research could incorporate tensor-train decomposed PINNs (TT-PINNs) on larger-scale and more sophisticated optical neural networks. Such advancements would pave the way for deploying massive neural networks within the constraints of optical hardware, providing a promising avenue for accelerating PDE solving capabilities in edge-computing environments.

In conclusion, this paper introduces a promising solution for real-time, on-chip PDE solving using optical neural networks. By eliminating the need for back propagation and leveraging the intrinsic benefits of photonic computing, this research lays the groundwork for innovative developments in the domain of physics-informed artificial intelligence.