Solving Inverse Problems With Physics-Informed DeepONet: A Practical Guide With Code Implementation | by Shuai Guo | Jul, 2023

Two case studies with parameter estimation and input function calibration

Photo by 愚木混株 cdd20 on Unsplash

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.

Figure 1. Operators transform one function into another, which is a concept frequently encountered in real-world dynamical systems. Operator learning essentially involves training a neural network model to approximate this underlying operator. A promising method to achieve that is DeepONet. (Image by author)

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.

Figure 2. In forward problems, the objective is to predict the outputs given the known inputs via the operator. In inverse problems, the process is reversed: known outputs are used to estimate the original, unknown inputs, often with only partial knowledge of the underlying operator. Both forward and inverse problems are commonly encountered in computational science and engineering. (Image by author)

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

Source link

Leave a Comment