Most Accurate/Efficient Interpolation approach for 1D/2D, discrete-continuous hybrid problem

Specific Domains Numerics
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Hi all, looking for advice on the best way to get very accurate results in finite computing time, for this problem:

A discrete grid in x, with f(x) values at each point. For any initial choice of x0, need to interpolate f(x) in the neighboring area, do a bunch of calculations on f(x) to get new function f2(x) in this neighborhood, then find where the local minimum of f2(x) is.
(The problem has to stay as continuous as possible, since the minimum place for f2, at xMin, does NOT have to be at one of the discrete data locations, but should be found as precisely as possible.)

But, this is actually a 2D problem, in time t and position x. Because after it calculates that xMin, it steps to xMin and ahead to the next time value, and then begins the process all over again for the next timestep: it finds the next xMin for the new t, and continues stepping again and again unless it gets to the end time. (At almost all times, the stepper will be located in between the actual discrete x data points, not exactly on one of them.)

Also, the problem is also somewhat continuous in t, because the whole 2D function I’m starting with is really obtained as a numerical solution to a partial differential equation using the Method of Lines. So, at each x point, what I really have isn’t just a single data point, but is an ODE interpolation solution x(t), so it is continuous in time (sort of). So, maybe there is a more holistic interpolation approach that can do the whole 2D interpolation in (t,x) at once, for the neighborhood around the place where the stepper is located at the moment.

This problem is complicated, so I’d appreciate any advice on either the simpler problem (for a single value of t) – discrete data points, interpolation, calculations with the interpolation, continuous minimum finding – or on the whole 2D problem (if possible).

Thanks!

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BTW, I’ve heard of ApproxFun for approximating a whole data set, but I’m not sure if that’s the best choice here, because I’m only interpolating a few (~10-20) f(x) data points in the neighborhood of my x for the stepper, not the function at all x everywhere.

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Your question is underspecified: a lot depends on the properties of your function.

I am not sure I fully understand the question, but a simple tensored spectral (eg Chebyshev polynomial) approach may work with sufficiently smooth/differentiable functions. As always, I would recommend

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A physics-informed neural network or a tensor product of 1D universal approximators as Tamas mentions.

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Thanks for the links & references guys, I’ll check it out.

For now, to develop some procedures, I’m just working with the plain wave equation:
ftt = fxx.
(That is, d^2/dt^2 {f(t,x)} = d^2/dx^2 {f(t,x)} .)

I guess what I was more specifically asking with my question, is whether Julia has a 2-dimensional version of a “spline”-like interpolator. Preferably where it can handle discrete data in one dimension (x), and 1D interpolation data (coming from ODE integrators) in the other diimension (t). Or, at least, a 2D interpolation that can handle discrete data in both dimensions (t,x).