The Piecewise Library

Identifying nonlinearities in mechanical systems from measurement data is a regression problem — but the basis functions matter. Polynomials can approximate smooth nonlinearities, but real mechanical systems have gaps, stops, and clearances that produce piecewise-linear restoring forces. Polynomials approximate these poorly.

arXiv:2603.16746 constructs a library of piecewise-linear springs with gaps, realized using min and max functions with biases. Each library element is a simple function: zero below a threshold, linear above it, or the reverse. The nonlinear restoring force is expressed as a linear combination of these elements, and the coefficients are found by solving a linear regression problem.

The library elements are the activation functions of the neural network framework, but the key insight is that they are physically interpretable — each one corresponds to a spring that engages at a specific displacement threshold. The identified model is not a black box. It is a collection of named mechanisms: this spring activates at 2mm, that one saturates at 5mm.

The method identifies a Duffing oscillator and a piecewise-linear oscillator with a gap from free-response data. It then validates on experimental data from a cantilevered plate under magnetic restoring force, accurately predicting steady-state responses under harmonic excitation.

The principle: if your nonlinearity has physical structure (thresholds, saturation, dead zones), build a library from elements that share that structure. The regression stays linear — you are finding coefficients, not fitting a neural network — but the basis functions carry the physics. The model inherits interpretability from the library, not from regularization or post-hoc analysis.