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Pre-built implicit layer architectures with O(1) backprop, GPUs, and stiff+non-stiff DE solvers, demonstrating scientific machine learning (SciML) and physics-informed machine learning methods

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PR adds Lagrangian NN implementation (#352) ![Screenshot from 2021-04-24 17-58-38](https://user-images.githubusercontent.com/41639271/115963832-d0c8e100-a529-11eb-80f4-300a8c99a994.png) Evident drawbacks: - For Hessian and Jacobians all cross-derivatives are computed before extracting the useful ones - Only one-point data...

Hi, I am trying to implement a working example for NeuralDAE. Since there is no example for NeuralDAE yet, so after looking at some issues, I chose the [test for...

The DiffEqFlux Layers need to satisfy https://lux.csail.mit.edu/dev/api/core/#Lux.AbstractExplicitLayer else the parameters/states returned from `Lux.setup` be incorrect. As pointed out in slack ``` julia> ps, st = Lux.setup(rng, Chain(node,Dense(2=>3))) ((layer_1 = NamedTuple(),...

Fixes https://github.com/SciML/DiffEqFlux.jl/issues/707

https://github.com/SciML/DiffEqFlux.jl/blob/e32422d86bea1cdda8797d0b909cb3e207738bd5/src/DiffEqFlux.jl#L24-L40 Related to https://github.com/SciML/SciMLSensitivity.jl/issues/582

Hi folks, I've been trying to use the two-stage training method for collocation (as described in this example https://diffeqflux.sciml.ai/stable/examples/collocation/). I can run the example itself and it works great, but...

This will allow doctesting with GPUs, and using the exclusive instances for building bigger examples. https://github.com/JuliaGPU/CUDA.jl/blob/master/.buildkite/pipeline.yml#L345-L364

This implements a forward-mode version of FFJORD via ForwardDiff. However, Dual tag ordering issues are showing up, so it's failing. ```julia using DiffEqFlux, DifferentialEquations, GalacticOptim, Distributions nn = Chain( Dense(1,...