Examples include electromagnetic modeling and thermal analysis of IC chips [1], medical imaging [2], safety verification of autonomous systems [3]. Discretization-based solvers (e.g., finite-difference and finite-element methods) convert a PDE to a large-scale algebraic equation via spatial and temporal discretization. Solving the resulting algebraic equation often requires massive digital resources and run times. Physics-informed neural network (PINN) is a promising discretization-free and unsupervised approach to solve PDEs [4]. PINN uses the residuals of a PDE operator and the boundary/initial conditions to set up a loss function, then minimizes the loss to train a neural network as a global approximation of the PDE solution. However, current PINN training typically needs several to dozens of hours on a powerful GPU, hindering the deployment of an real-time neural PDE solver on edge devices.

Solving partial differential equations (PDEs) numerically often requires huge computing time, energy cost, and hardware resources in practical applications. This has limited their applications in many scenarios (e.g., autonomous systems, supersonic flows) that have a limited energy budget and require near real-time response. Leveraging optical computing, this paper develops an on-chip training framework for physics-informed neural networks (PINNs), aiming to solve high-dimensional PDEs with fJ/MAC photonic power consumption and ultra-low latency. Despite the ultra-high speed of optical neural networks, training a PINN on an optical chip is hard due to (1) the large size of photonic devices, and (2) the lack of scalable optical memory devices to store the intermediate results of back-propagation (BP). To enable realistic optical PINN training, this paper presents a scalable method to avoid the BP process. We also employ a tensor-compressed approach to improve the convergence and scalability of our optical PINN training. This training framework is designed with tensorized optical neural networks (TONN) for scalable inference acceleration and MZI phase-domain tuning for \textit{in-situ} optimization. Our simulation results of a 20-dim HJB PDE show that our photonic accelerator can reduce the number of MZIs by a factor of $1.17\times 10^3$, with only $1.36$ J and $1.15$ s to solve this equation. This is the first real-size optical PINN training framework that can be applied to solve high-dimensional PDEs.

Moreover, environmental changes due to temperature or mechanical strains can lead to time-varying effects which require adaptive equalization. Adaptive equalizers are indeed commonplace in optical receivers [1, 2], typically implemented via gradient-descent-based least-mean squares filtering [3]. For example, in coherent systems such equalizers can track the inverse Jones matrix of the channel and may also correct for additional distortions such as residual chromatic dispersion [4]. However, the underlying equalizer structure is linear, which limits the type of functionalities that can be expressed and therefore also the performance that can be achieved. To overcome the limitations of linear equalizers, a wide variety of machine learning (ML) algorithms have recently been proposed and verified in hardware (HW). For example, field-programmable gate array (FPGA) implementations of various neural network equalizers were demonstrated for IM/DD links [5], passive optical networks [6], optical interconnects [7], and coherent systems [8].