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!

## Table of Content

· 1. PI-DeepONet: A refresher

· 2. Problem Statements

· 3. Problem 1: Parameter Estimation

∘ 3.1 How it works

∘ 3.2 Implementing a…