In my previous blog, we delved into the concept of physics-informed DeepONet (PI-DeepONet) and explored why it is particularly suitable for operator learning, i.e., learning mappings from an input function to an output function. We also turned theory into code and implemented a PI-DeepONet that can accurately solve an ordinary differential equation (ODE) even with unseen input forcing profiles.
The ability to solve these forward problems with PI-DeepONet is certainly valuable. But is that all PI-DeepONet can do? Well, definitely not!
Another important problem category we frequently encountered in computational science and engineering is the so-called inverse problem. In essence, this type of problem reverses the flow of information from output to input: the input is unknown and the output is observable, and the task is to estimate the unknown input from the observed output.
As you might have guessed, PI-DeepONet can also be a super useful tool for tackling these types of problems. In this blog, we will take a close look at how that can be achieved. More concretely, we will address two case studies: one with parameter estimation, and the other one with input function calibrations.
This blog intends to be self-contained, with only a brief discussion on the basics of physics-informed (PI-) learning, DeepONet, as well as our main focus, PI-DeepONet. For a more comprehensive intro to those topics, feel free to check out my previous blog.
With that in mind, let’s get started!