Fig. 1. Setup of photonic chip for solving PDEs. (a) The setup of the photonic solver. Light from the optical frequency comb is separated by a DWDM to provide nine wavelength channels. A set of VOAs is used to modulate the light intensity of the wavelength channels, which represent the value of the input vector. The modulated light is injected into the chip through the fiber array. The chip consists of an inversely designed 1:9 power splitter and a 9×9 microring array, which loads the input matrix by tuning the resonant wavelength of each microring through electrodes. The result of MVM is collected by a set of power meters and stored in an external cache. The controller processes the data stored and controls the solving process. (b) The microscope image of the fabricated photonic chip bonded on a PCB, which contains two solvers. (c) Top panel: horizontal silicon waveguides and vertical metal wires. The left bottom panel shows the details of the microring array. Right bottom image: white color denotes silicon waveguides. Each microring is covered by a TiN electrode denoted with a dark green color. The yellow color denotes the layer of AlCu, which is used to conduct current.

Fig. 2. Sketch diagram of MVM for PDE solving. (a) First, the PDE is converted into matrix form using the finite difference method. The resulting coefficient matrix is partitioned into blocks, which are then loaded onto photonic chips that act as MVM cores during the solving process. The input vector is divided into slices and sequentially loaded onto different wavelength channels, where the light signals are separated by a DWDM. Utilizing the microring arrays and power splitters on the photonic chips, these input vector slices are processed in parallel across the wavelength channels. By controlling the transmission characteristics of the microrings, MVM is achieved. Finally, the results are collected and stored by photodetectors, enabling efficient and accurate PDE solving. The different colors of the squares in the figure represent different values. (b) The blue line represents the measured output signal by cascading optical frequency comb, DWDM, and VOAs. Each signal here can be tuned independently to the load vector element. The red line represents the measured spectrum of the test ring of the photonic chip. The free spectrum range is 15.78 nm, which covers the signal channels. (c) The measured movement of the resonant wavelength when the different voltages are added. (d) The top left part of the coefficient matrix of the Laplacian operator with a mesh size of 6 × 6. Matrix patches are marked by red squares. (e) The measured normalized spectra of nine rows of the microring array with nine DWDM wavelength channels opened. The central wavelengths of nine channels are marked with arrows at the top of the figure. (f) The measured normalized power matrix of the microring array.

Fig. 3. Solving results of time-evolving PDEs. The results of solving different kinds of time-evolving PDEs are presented here. (a)–(d) The solving results of the heat equation with a mesh size of 6×6; the heat equation is shown at the top. (a) The evolution process of the solution at grid (3,3); (b) the overall accuracy of the solution at different times in the evolution process; the accuracy is more than 95%. (c), (d) The comparison of the experimental results and simulation results at the time points of 0.25, 1.25, and 3.75. (e)–(h) The solving results of the wave equation with a mesh size of 6×6; the wave equation is shown at the top. (e) The evolution process of the solution at grid (3,3); (f) the overall accuracy of the solution at different times in the evolution process; the accuracy is more than 85%. (g), (h) The experimental and simulation results at the time points of 0.05, 0.40, and 0.9. (i)–(l) The solving results of the Burgers equation with a mesh size of 9×9; the Burgers equation is shown at the top. (i) The evolution process of the solution at grid (2,8); (j) the overall accuracy of the solution at different times in the evolution process; the accuracy is more than 94%. (k), (l) The experimental and simulation results at the time points of 0.20, 0.80, and 1.40. Sim., simulation; Exp., experiment.

Fig. 4. Solving results of the Poisson equation. Here, the Poisson equation is solved with fine mesh and high accuracy. The expression of the Poisson equation is given at the top. (a), (b) Simulation and experimental results of the Poisson equation, which agree well with each other. The mesh size is 12×12, which is the highest report in photonic computing of PDEs. (c) The error distribution with an average error of 6.5%; (d) the statistical distribution of the accuracy of the solution at the 144 (12×12) grid points. The average accuracy of the experimental results is 93.5%, with a standard deviation of 0.0507. Sim., simulation; Exp., experiment.

Fig. 5. Parallel solving results of PDEs. Two PDEs are solved in parallel on a single photonic computing platform. (a) The top left part of coefficient matrixes of two PDEs with a mesh of 6×6. (b) The measured normalized spectra of nine rows of the microring array with matrix patches of two PDEs loaded on chip. (c) The measured normalized power matrix of the microring array. (d) 3D visualization of Laplace’s equation with trigonometric function boundary conditions. The mesh is set as 9×9. (e) Comparison of simulation and experimental results of Laplace’s equation. (f) The error distribution of Laplace’s equation with an average error of 4.1%. (g) 3D visualization of the Poisson equation with a kind of linear function boundary condition. (h) Comparison of simulation and experimental results of the Poisson equation. (i) The error distribution of the Poisson equation, with an average error of 4.2%.