Integro-differential equation PINN solver benchmark

[killah-t-cell]

Would be great to have a benchmark for IDE PINNs.

[killah-t-cell]

Specially since IDEs are much slower than PDEs in NeuralPDE (training sometimes goes from 1 minute to 1 hour by adding one integral). This is to be expected as we discussed, but it would be cool to have a benchmark as a way of checking speed and accuracy against other methods.

[Spinachboul]

@killah-t-cell
Then it would require benchmarks which have some kind of noise factor.
But we cant have muti-dimensional problems

[ArpanC6]

Hi @killah-t-cell @ChrisRackauckas,

I'd like to take on this benchmark. I have experience with SciMLBenchmarks.jl (PRs #1552, #1554, #1569 on Bayesian inference) and numerical methods research (JAX/PyTorch AD analysis at Cornell).

Proposed benchmark scope:

Problem: 1D Volterra integro-differential equation from the NeuralPDE tutorial:
∂u/∂t + 2u + 5∫₀ᵗ u(x)dx = 1, u(0) = 0, t ∈ [0, 2]

Exact solution: u(t) = (1/2) * e^(-t) * sin(2t)

Methods compared:

1. PINN (NeuralPDE.jl) - Physics-informed neural network with training time and convergence metrics
2. Traditional - Convert IDE to ODE system via differentiation + solve with Tsit5/Rodas5

Metrics:

• Training/solve time vs accuracy (L2 error)
• Convergence iterations vs error
• Memory usage during training
• Scaling: compare with pure ODE equivalent (same problem without integral term)

Output: .jmd benchmark with work precision style diagrams

Questions:

• Should I include Fredholm-type IDE as well, or start with Volterra only?
• Is there a preferred traditional solver to benchmark against (MethodOfLines, direct quadrature+ODE, or both)?

I'll start prototyping with the NeuralPDE IDE tutorial and aim for a draft PR in 2 - 3 weeks.

Thanks.

[ChrisRackauckas]

okay cool yeah we need this.