Improve Parsing with Symbolics

[xtalax]

The current setup was written before we had all the useful tools present in Symbolics.jl, this PR takes advantage of these and the PDE helper functions present in PDEBase

[YichengDWu]

I recently completely rewrote the parsing in YichengDWu/Sophon.jl#204, and it is now close to the ideal state I mentioned in #687. I used pattern matching from MacroTools there to transform expressions. Although I don't understand what you are doing here, I guess converting it to Symbolics-based rewriting would be straightforward. I hope this helps with this PR.

[YichengDWu]

It should just be a single directional vector? get_ε(w). Mixed derivatives are handles outside of the derivative function.

[xtalax]

Just trying to find a way to do this without having to generate rules for every combination of variables

[xtalax]

@ChrisRackauckas @YichengDWu
Ok, here is what I expect will work, I'd appreciate a review and perhaps a comment on how mixed deriv rules can be generated without being exhaustive. @YichengDWu can you clarify what needs to change for mixed derivs to work as in your PR?

We have lost functionality for multioutput and integrals, to be re-added in future PRs, is this acceptable?

[YichengDWu]

I'm not sure if I understand

@rule $(Differential(~x)^(~d::isinteger)(~w)) => derivative(phi, u, ~x, get_εs(~w), ~d, θ)

It looks like we should pass the coordinate to derivative not just ~x here? Anyway, mimicking this rule, mixed derivatives can be handle like the following (but not exactly):

@rule $(Differential(~x1)^(~d1::isinteger)($(Differential(~x2)^(~d2::isinteger)(~w))))
=>  derivative(phi, (cord_, θ_, phi_) ->derivative(phi_, u, ~x2, get_ε(~w), ~d2, θ_), u, ~x1, get_ε(~w), ~d1, θ)

Can you symbolic_discretize some test PDE to inspect the generated expression?

[YichengDWu]

We have lost functionality for multioutput and integrals, to be re-added in future PRs, is this acceptable?

Or move the code into a submodule, and use Preferences to switch between the old and new backends, until the day when the old parsing backend is completely replaced?

[xtalax]

This changes fundamentally a large part of the codebase, it will add a lot of complexity to do this, I'd rather keep them on different major versions instead

[ChrisRackauckas]

If anyone needs something backported to the major that's fine, but I'd like to just out with the old and in with the new. On a major it's fine to be breaking, and I think if we have a functionality regression for a major change in the maintainership of this package it's fine. Honestly the integro-differential equation piece needs a bit more though on part of the definition anyways, so we should come back to it but I wouldn't let that hold it up. And the multioutput is a very minor feature that I wouldn't say is an important one.

[sathvikbhagavan]

The loss functions generated are incorrect.

Running this example:

@parameters θ
@variables u(..)
Dθ = Differential(θ)

# 1D ODE
eq = Dθ(u(θ)) ~ θ^3 + 2 * θ + (θ^2) * ((1 + 3 * (θ^2)) / (1 + θ + (θ^3))) -
                u(θ) * (θ + ((1 + 3 * (θ^2)) / (1 + θ + θ^3)))

# Initial and boundary conditions
bcs = [u(0.0) ~ 1.0]

# Space and time domains
domains = [θ ∈ Interval(0.0, 1.0)]

# Neural network
chain = Lux.Chain(Lux.Dense(1, 12, Flux.σ), Lux.Dense(12, 1))

discretization = NeuralPDE.PhysicsInformedNN(chain, GridTraining(0.1))

@named pdesys = PDESystem(eq, bcs, domains, [θ], [u(θ)])
prob = NeuralPDE.discretize(pdesys, discretization)
prob.f.f(prob.u0, nothing) # Calculate loss

Calculating the loss is erroring out.

I am getting Runtime Generated Function in https://github.com/xtalax/NeuralPDE.jl/blob/parsing/src/loss_function_generation.jl#L127 as

ex = :(function (var"##arg#8771009302963811065", θ_SYMBOL, phi, var"##arg#5964805160111424296")
      #= /home/sathvikbhagavan/.julia/packages/SymbolicUtils/ssQsQ/src/code.jl:373 =#
      #= /home/sathvikbhagavan/.julia/packages/SymbolicUtils/ssQsQ/src/code.jl:374 =#
      #= /home/sathvikbhagavan/.julia/packages/SymbolicUtils/ssQsQ/src/code.jl:375 =#
      begin
          θ = var"##arg#8771009302963811065"[1]
          nothing
      end
  end)

which does not look correct. I inserted show statements in https://github.com/xtalax/NeuralPDE.jl/blob/parsing/src/loss_function_generation.jl#L117 and https://github.com/xtalax/NeuralPDE.jl/blob/parsing/src/loss_function_generation.jl#L118 and found:

expr = 82570.18592591035(-phi(SymbolicUtils.BasicSymbolic{Real}[-6.055454452393343e-6 + θ], θ_SYMBOL) + phi(SymbolicUtils.BasicSymbolic{Real}[6.055454452393343e-6 + θ], θ_SYMBOL)) + ((-1 - 3(θ^2))*(θ^2)) / (1 + θ + θ^3) - 2θ + ((1 + 3(θ^2)) / (1 + θ + θ^3) + θ)*phi(SymbolicUtils.BasicSymbolic{Real}[θ], θ_SYMBOL) - (θ^3)
expr = nothing

So doing expr = swch(expr) in https://github.com/xtalax/NeuralPDE.jl/blob/parsing/src/loss_function_generation.jl#L118 is returning nothing as there is no switch in the first expression.

So, there might be a bug in how the symbolic rules are written. I am not familiar with SymbolicUtils that much to point out the bug.

@xtalax

cc: @ChrisRackauckas

[xtalax]

Try wrapping the rule in a chain before postwalk, and maybe also a vector would be my guess