  Numerical integration is a foundational technique in scientific computing and
is at the core of many computer vision applications. Among these applications,
neural volume rendering has recently been proposed as a new paradigm for view
synthesis, achieving photorealistic image quality. However, a fundamental
obstacle to making these methods practical is the extreme computational and
memory requirements caused by the required volume integrations along the
rendered rays during training and inference. Millions of rays, each requiring
hundreds of forward passes through a neural network are needed to approximate
those integrations with Monte Carlo sampling. Here, we propose automatic
integration, a new framework for learning efficient, closed-form solutions to
integrals using coordinate-based neural networks. For training, we instantiate
the computational graph corresponding to the derivative of the network. The
graph is fitted to the signal to integrate. After optimization, we reassemble
the graph to obtain a network that represents the antiderivative. By the
fundamental theorem of calculus, this enables the calculation of any definite
integral in two evaluations of the network. Applying this approach to neural
rendering, we improve a tradeoff between rendering speed and image quality:
improving render times by greater than 10 times with a tradeoff of slightly
reduced image quality.
