Radius-selection conjecture for weak-form system identification

Let $\mathbf{r}(\mathbf{u},\mathbf{w})$ be the residual in the weak-form system-identification method, and let $\mathbf{e}_{\text{int}}(r)$ denote the numerical integration error as a function of the test-function support radius $r$. Let $r_c$ be the point at which the monotonic decrease of $\log(\lVert\mathbf{e}_{\text{int}}(r)\rVert)$ stagnates. Assume that the residual is zero-mean:

$$\mathbb{E}[\mathbf{r}(\mathbf{u},\mathbf{w})]=\mathbf{0}.$$

Radius-selection conjecture. The stagnation radius satisfies

$$r_c \equiv \arg\min_r \mathbb{E}\left[\left\lVert\widehat{\mathbf{w}}(r)-\mathbf{w}^{\star}\right\rVert_2\right],$$

where $\widehat{\mathbf{w}}(r)$ is the WENDy estimator with test-function radius $r$. The conjecture formalizes the proposed data-driven choice of test-function radius: the integration error initially decreases with $r$, and its stagnation point is expected to coincide with the radius minimizing the expected parameter-estimation error.