AbstractThis research proposes a deep-learning paradigm, termed functional learning (FL), to physically train a loose neuron array, a group of non-handcrafted, non-differentiable, and loosely connected physical neurons whose connections and gradients are beyond explicit expression. The paradigm targets training non-differentiable hardware, and therefore solves many interdisciplinary challenges at once: the precise modeling and control of high-dimensional systems, the on-site calibration of multimodal hardware imperfectness, and the end-to-end training of non-differentiable and modeless physical neurons through implicit gradient propagation. It offers a methodology to build hardware without handcrafted design, strict fabrication, and precise assembling, thus forging paths for hardware design, chip manufacturing, physical neuron training, and system control. In addition, the functional learning paradigm is numerically and physically verified with an original light field neural network (LFNN). It realizes a programmable incoherent optical neural network, a well-known challenge that delivers light-speed, high-bandwidth, and power-efficient neural network inference via processing parallel visible light signals in the free space. As a promising supplement to existing power- and bandwidth-constrained digital neural networks, light field neural network has various potential applications: brain-inspired optical computation, high-bandwidth power-efficient neural network inference, and light-speed programmable lens/displays/detectors that operate in visible light.