Derivative-based SINDy (DSINDy): Addressing the Challenge of Discovering Governing Equations from Noisy Data

The paper’s Theorem 5.4 gives the PSDN relative-error bound with constants C1 and C2 under rank and small-perturbation assumptions, using (i) a triangle-inequality decomposition, (ii) a projector perturbation bound depending on the design-matrix perturbation, (iii) unbiased-library variance control through the trapezoidal integrator, and (iv) a trapezoid-rule quadrature error. The candidate solution reproduces the same decomposition, uses a standard subspace-perturbation inequality (equivalent up to constants to the paper’s Lemma SM3.8), derives the same C2 from the same variance bound for TΔΘ, and invokes the same trapezoidal error rate for the antiderivative. It arrives at exactly the stated inequality and the same asymptotics. Minor differences (e.g., a direct Stewart–Sun bound versus the paper’s SM3.6→SM3.8 route, and the explicit smallness condition ‖(Φ*)†‖‖ΔΦ‖<1/4) do not change the result.