Solving differential equations with Neural Networks

[@haldanesghost]

Gonna be jumping on the bandwagon and proceed to give you my most sincere form of gratitude for these lectures. As a researcher with a foot in many fields at once, there comes a point when one can’t just keep up with every advance in every field one is paying attention to by just reading the literature or half-baked documentation. This is where skillful teaching really shines. I’ve only begun the lectures, but look forward to combing through them-- especially the PDE part. Cheers from the Caribbean 🏝️

[@MortenHjorthJensen]

This is exactly what I was hoping for. Your comments really warm and feel free to use the material for various courses as you want, see github.com/mhjensen for more educational material. Sorry for the late response, I had overseen several mails from youtube since I did not have a warning on.

[@moosagaming7519]

I would like to pay my sincere thanks to Professor for your valuable lectures on Neural Network to solve differential equations. They helped me a lot in solving boundary layer problems

[@MortenHjorthJensen]

Thank you so much for your kind words. Best wishes

[@tiamiyuabdgafar6909]

Thank you so much for this incredible lecture. Is there a TensorFlow or PyTorch version of these codes?

[@MortenHjorthJensen]

Thx so much for your kind comments. I will upload a tensor flow version as soon as possible. Feel free to send me an email to [e-mail] and I will send you a Jupyter-notebook.

[@himanshugaur9764]

Dear sir, Thank you very much for the valuable Video. I was looking for these examples since a long time.

[@MortenHjorthJensen]

Thank you very much and best wishes

[@dondocentone1334]

Amazing lecture! I just wonder - when I have a nonzero source term on the right hand side of the ODE which is a function of X (for example shaft torque difference in jet engine RPM governing equation being function of time) and the source term depends on other variables (fuel flow, temperature, pressure), should the inputs od ANN include the other variables alongside with X variable? Becouse as far as I studied PINNs, they are used for self-evolving dynamic systems (zero or constant source terms, closed systems, Navier-Stokes, diffusion, convection etc.), where all that is needed to train the ANN consist of Y training data and outputs of the ANN itself (autograd derivatives and so on) and there is no need to use other measured data except the one output variable Y. But in the case of jet engine the right hand side - acceleration rate - is a function of time which must be known from data (measured or computed) in order to estimate ODE cost. Maybe I'm missing something, or can't PINNs really be used for modelling open thermodynamic systems? Thanks, Professor. EDIT: I mean the case, where the RHS is not only function of Y(X) but also other variables which are functions of X

[@nagari_ishere]

Wounderfull

[@MortenHjorthJensen]

Thank you so much!

[@MortenHjorthJensen]

Thx so much, and feel free to look up more at my github.com/mhjensen address. Best wishes

[@kasahun86]

Thank you for it ! can i get the code for advection equation using neural network?

[@MortenHjorthJensen]

Thx so much, and feel free to look up more at my github.com/mhjensen address. Best wishes and sorry for the late response, I had overseen several mails from youtube since I did not have a warning on. The codes are at github.com/CompPhysics/MachineLearning/blob/master/doc/pub/week43/ipynb/week43.ipynb

[@MariaHeger-tb6cv]

What about trying this on more difficult equations, e.g. Navier-Stokes??

[@MortenHjorthJensen]

Indeed, this is the plan, however, keep in mind that the present course is at the senior undergrad level and/or beginning master of science level, 3rd or 4th year of study and many students are not fully familiar with PDEs. For my advanced course, FYS5429 at the university of Oslo, the plan is indeed to do so, but then in connection with Physics informedn NN. You can find an example of such a project, with links to codes etc at github.com/CompPhysics/AdvancedMachineLearning/blob/main/doc/Projects/2023/ProjectExamples/PINNs.pdf

[@abdessamadmehdari6914]

What about CNN

[@MortenHjorthJensen]

Thx for the comment. Are you thinking of CNNs applied to the solution of differential equations? Else, if you are interested in solving differential equations with ML, I recommend strongly Brunton's and Kutz' recent text on data driven science and engineering.

[@ClosiusBeg]

Yes, for small equations it is working fine, but what about ODEs of CFD or FEA?

[@MortenHjorthJensen]

I would strongly recommend Steve Brunton's and Kutz' work on this, see their textbook and also a video here, kzbin.info/www/bejne/f4mwhIKJes9karM. This is indeed a very active and interesting research field. Highly recommended. They have, with colleagues, really been pioneering work on such topics.