A Appendix

We will prove by the induction. Let's suppose that the formula holds for By the definition in Eq. 4 and the chain rule, we can get that: N In this section, we give error bounds for spline representation. In the present work, we focus on using spline for smoothing noisy data. Following [51], we have spline fitting error bounds, as following. Eq. 11 L Output: Mean estimation: θ A.4 Training Details Additional training hyper parameters used in Sec. 4 is shown in the Tab. 2. T able 2: Training Details We list additional discovery and UQ results in this section.