When solving, modeling or reasoning about complex problems, it is usually convenient to use the knowledge of a parallel physical system for representing it. This is the case of lumped-circuit abstraction, which can be used for representing mechanical and acoustic systems, thermal and heat-diffusion problems and in general partial differential equations. Integrated photonic platforms hold the prospective to perform signal processing and analog computing inherently, by mapping into hardware specific operations which relies on the wave-nature of their signals, without trusting on logic gates and digital states like electronics. Here, we argue that in absence of a straightforward parallelism a homomorphism can be induced. We introduce a photonic platform capable of mimicking Kirchhoff’s law in photonics and used as node of a finite difference mesh for solving partial differential equation using monochromatic light in the telecommunication wavelength. Our approach experimentally demonstrates an arbitrary set of boundary conditions, generating a one-shot discrete solution of a Laplace partial differential equation, with an accuracy above 95% with respect to commercial solvers. Our photonic engine can provide a route to achieve chip-scale, fast (10 s of ps), and integrable reprogrammable accelerators for the next generation hybrid high-performance computing. Summary A photonic integrated platform which can mimic Kirchhoff’s law in photonics is used for approximately solve partial differential equations noniteratively using light, with high throughput and low-energy levels.

中文翻译：

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诱导同态：光子学中的基尔霍夫定律

在解决，建模或对复杂问题进行推理时，通常使用并行物理系统的知识来表示它很方便。集总电路抽象就是这种情况，可用于表示机械和声学系统，热和热扩散问题以及一般的偏微分方程。集成的光子平台通过映射到依赖于其信号波形的特定于硬件的操作中而无需依靠逻辑门和电子设备等数字状态，从而有望在本质上执行信号处理和模拟计算。在这里，我们认为，在没有直接并行性的情况下，可以诱发同构。我们介绍了一种能够模仿光子学中的基尔霍夫定律的光子平台，并用作有限差分网格的节点，用于使用电信波长的单色光求解偏微分方程。我们的方法通过实验证明了任意一组边界条件，生成了Laplace偏微分方程的单次离散解，相对于商用求解器，其精确度高于95％。我们的光子引擎可以为下一代混合高性能计算提供一种实现芯片级，快速（10 ps ps）和可集成可重编程加速器的途径。小结可以模拟光子学中的基尔霍夫定律的光子集成平台，用于非迭代地使用光近似求解偏微分方程，