Testing derivatives via Integral

[freemin7]

The fundamental theorem of calculus says that differential problems can also be expressed as Integral problems. Certain functions like clamp, max, min lack a definition of derivatives at every point. When you perform a finite difference approximation the result will certainly be in doubt around these points. Integrals would be mostly immune against those defects.

Do you have any thoughts on idea?

[willtebbutt]

I would be more inclined to suggest using either forwards- or reverse- differencing around discontinuities. i.e you set up your tests so that finite differences avoids the discontinuities. See eg. FiniteDifferences.forward_fdm and FiniteDifferences.reverse_fdm.

While I agree that quadrature-based tests are appealing, I would imagine they'll also struggle in the vicinity of discontinuities since they usually assume some form of smoothness.

[oxinabox]

It's an interesting idea. But I feel like it would be hard to get accuracy. Also really expensive at higher dimensions? Not sure.

Testing both sides of discontinualities is I think pretty standard.

Would be interesting to see it coded up and tested. Even if it doesn't work great, would still make an interesting blog post