Physics informed neural operator ode

[KirillZubov]

Implementation Physics-informed neural operator method for solve parametric Ordinary Differential Equations (ODE) use DeepOnet.

#575

Checklist

• pino ode
• family ode by parameter
• physics informed DeepOnet
• tests
• addition loss test
• doc
• multiple parameters
• test with vector outputs and multiple parameters
• imigrate to LuxNeuralOperators
• interpretation output with another mesh
• vector output Vector output for PINO ODE #871
• update docs

https://arxiv.org/abs/2103.10974
https://arxiv.org/abs/2111.03794

[KirillZubov]

@ChrisRackauckas I need help with packages version. Adding dependency of NeuralOperator.jl to project, fail CI. I've tried a little to line up suitable versions but not success.

[sathvikbhagavan]

@KirillZubov, This looks good to go, but I do not want to merge until CI is ran successfully. So, have to wait a bit on that. Until then, I see Chris has pinged you to look into #883. Can you look into fixing that?

[KirillZubov]

@KirillZubov, This looks good to go, but I do not want to merge until CI is ran successfully. So, have to wait a bit on that. Until then, I see Chris has pinged you to look into #883. Can you look into fixing that?

Hey @sathvikbhagavan yes, it is related to my task with output vector so yes I going to figure out to it on this week.
I agree there will be needed sure that all work fine before merge it. I will complete this task, waiting for the remaining parts to be ready in order to merge this PR.

[KirillZubov]

Hi @sathvikbhagavan, @ChrisRackauckas I updated this PR for Lux v1.0 and some another update in order to PinoODE tests would passed for CI. The fact that other tests don't work due to the Lux update is beyond the scope of my summer project. This PR is ready for review.