Index Reduction

torchdae's Index Reduction supports differentiation and dummy_derivative modes.

To start with the Index Reduction, we run the simplify_dae along with the desired mode.

The function will return the function wrapped in either IndexReducedDAE or DummyDerivativeDAE depending on the mode along with the current state y0 and its derivative yp0.

Internally, the simplify_dae function runs the analyze_pytorch_dae function to get the DAEStructure that contains equations and properties of the input function.

The differentiation takes the derivative of the equation depending on the differentiation_orders of the inputted equations using the Pantelides' algorithm and implements Baumgarte stabilization for higher index equations, requiring an alpha parameter.

The dummy_derivative mode implements the Mattsson-Söderlind algorithm without requiring an alpha parameter.

An example with simplify_dae:

import torch
from torchdae import simplify_dae, solve_bdf1

# Index-3 pendulum system
# (Positions x, y constraint: x^2 + y^2 = L^2)
# x^2 + y^2 - L^2 = 0
def pendulum_physics(t, y, yp):
    # ... (Pendulum equations of motion) ...
    ...

t0 = 0.0
# y0 = [x, y, vx, vy, lambda]
y0 = torch.tensor([[1.0, 0.0, 0.0, 0.0, 0.0]], dtype=torch.float64)
yp0_guess = torch.tensor([[0.0, 0.0, 0.0, -9.81, 0.0]], dtype=torch.float64)

F_reduced, Z0, Zp0 = simplify_dae(
    F=pendulum_physics,
    t0=t0,
    y0=y0,
    yp0=yp0_guess,
    algorithm="dummy_derivative"
)

sol = solve_bdf1(
    F=F_reduced,
    t_span=(t0, 2.0),
    y0=Z0,
    h=0.01
)