Operator Learning via Physics-Informed DeepONet: Let’s Implement It From Scratch

Ordinary and partial differential equations (ODEs/PDEs) are the backbone of many disciplines in science and engineering, from physics and biology to economics and climate science. They are fundamental tools used to describe physical systems and processes, capturing the continuous change of quantities over time and space. Yet, a unique trait of many of these equations is that they don’t just take single values as inputs, they take functions. For example, consider the case of predicting vibrations in a building due to an earthquake. The shaking of the ground, which varies over time, can be represented as a function that acts as the input to the differential equation describing the building’s motion. Similarly, in the case of sound waves propagating in a concert hall, the sound waves produced by a musical instrument can be an input function with varying volume and pitch over time. These varying input functions fundamentally influence the resulting output functions — the building’s vibrations and the acoustic field in the hall, respectively. Traditionally, these ODEs/PDEs are tackled using numerical solvers like finite difference or finite element methods. However, these methods come with a bottleneck: for every new input function, the solver must be run all over again. This process can be computationally intensive and slow, particularly for intricate systems or high-dimensional inputs. <a href="https://towardsdatascience.com/operator-learning-via-physics-informed-deeponet-lets-implement-it-from-scratch-6659f3179887">Learn More</a>