Fitting Second-Order Systems (e.g., Duffing Oscillator) – How to Handle First-Order Variables?

[abelleipardo]

I would like to fit a second-order dynamical system, specifically something like the Duffing oscillator. When written in first-order form, the full system is:
xdot = X
ydot = Y
Xdot = fcn1(x,y)
Ydot = fcn2(x,y)

In the paper "A Unified Sparse Optimization Framework to Learn Parsimonious Physics-Informed Models From Data" by Champion et al. (2020), the authors state:
“Because the variables X and Y are fixed to represent momentum, we apply SINDy only to fit the second two equations.”

I would like some clarification on how to implement this in practice. Specifically:

• Should I manually compute xdot and ydot from my simulation data, i.e., treat X and Y as known (from the simulation), and then only fit Xdot and Ydot using SINDy?
• If so, how should I prepare the data for SINDy so it understands which variables to treat as known and which to fit?
• I’m using the ConstrainedSR3 optimizer—are there any special considerations when applying it in this context?

Thank you for your help!

[Jacob-Stevens-Haas]

Hey, thanks for your question!

• You have options, but yes, you need to manually compute X and Y (potentially using pysindy.differentiation). You could fit all four variables (x, X, y, Y) and let SINDy discover the first two equations, but to do as Champion et al, you need to only fit Xdot and Ydot
• To do this, you would pass in X and Y as your input variable. Then you would pass in x and y as control variables. To wit:

model.fit(momentum, u=position)

where momentum is an array containing X and Y stacked along the last axis and position comprises x and y stacked likewise.

• Nope, not that I can think of. Two things to know with writing constraints:

1. you need to know the function library ordering, so it helps to use feature_lib.fit(arr).get_feature_names() before writing constraints.
2. Constraint format has changed between 1.7.5 and 2.0.0, so check the documentation of whatever version is installed. It may not line up with how the examples create constraints.