Neural networks can be used as a method for efficiently solving difficult partial differential equations. Efficient implementations of physics-informed machine learning from recent papers are being explored as part of the NeuralPDE.jl package. The issue tracker contains links to papers which would be interesting new neural network based methods to implement and benchmark against classical techniques.
Recommended Skills: Background knowledge in numerical analysis and machine learning.
Expected Results: New neural network based solver methods.
Mentors: Chris Rackauckas and Sathvik Bhagavan
Expected Project Size: 175 hour or 350 hour depending on the chosen subtasks.
Difficulty: Easy to Hard depending on the chosen subtasks.
Neural ordinary differential equations have been shown to be a way to use machine learning to learn differential equation models. Further improvements to the methodology, like universal differential equations have incorporated physical and biological knowledge into the system in order to make it a data and compute efficient learning method. However, there are many computational aspects left to explore. The purpose of this project is to enhance the universal differential equation approximation abilities of DiffEqFlux.jl, adding features like:
Improved adjoints for DAEs and SDEs
Various improvements to minibatching
Support for second order ODEs (i.e. symplectic integrators)
See the DiffEqFlux.jl issue tracker for full details.
This project is good for both software engineers interested in the field of scientific machine learning and those students who are interested in perusing graduate research in the field.
Recommended Skills: Background knowledge in numerical analysis and machine learning.
Expected Results: New and improved methods for neural and universal differential equations.
Mentors: Chris Rackauckas and Anas Abdelrehim
Expected Project Size: 175 hour or 350 hour depending on the chosen subtasks.
Difficulty: Medium to Hard depending on the chosen subtasks.
In many cases, when attempting to optimize a function f(p) each calculation of f is very expensive. For example, evaluating f may require solving a PDE or other applications of complex linear algebra. Thus, instead of always directly evaluating f, one can develop a surrogate model g which is approximately f by training on previous data collected from f evaluations. This technique of using a trained surrogate in place of the real function is called surrogate optimization and mixes techniques from machine learning to accelerate optimization.
The purpose of this project is to further improve Surrogates.jl by: adding new surrogate models, adding new optimization techniques, showcasing compatibility with the SciML ecosystem and fixing unwanted behaviour with some current surrogate models. The issue tracker contains list of new surrogate models which can be added.
Recommended Skills: Background knowledge of standard machine learning, statistical, or optimization techniques. Strong knowledge of numerical analysis is helpful but not required.
Expected Results: Improving Surrogates.jl with new surrogate models and new optimization techniques.
Mentors: Chris Rackauckas and Sathvik Bhagavan
Expected Project Size: 175 hour or 350 hour depending on the chosen subtasks.
Difficulty: Medium to Hard depending on the chosen subtasks.
Global Sensitivity Analysis is a popular tool to assess the effect that parameters have on a differential equation model. A good introduction can be found in this thesis. Global Sensitivity Analysis tools can be much more efficient than Local Sensitivity Analysis tools, and give a better view of how parameters affect the model in a more general sense. The goal of this project would be to implement more global sensitivity analysis methods like the eFAST method into GlobalSensitivity.jl which can be used with any differential equation solver on the common interface.
Recommended Skills: An understanding of how to use DifferentialEquations.jl to solve equations.
Expected Results: Efficient functions for performing global sensitivity analysis.
Mentors: Chris Rackauckas and Vaibhav Dixit
Expected Project Size: 175 hour or 350 hour depending on the chosen subtasks.
Difficulty: Easy to Medium depending on the chosen subtasks.