MMP.17:1 - Problem frame

Use this when repeated use of a model is too costly or cumbersome, and the next calculation needs only some of its responses. You may need values at many inputs, a probability of exceeding a limit, a field over a region, or a response used inside an inverse problem. Begin with one receiving question: which response must the replacement supply, over which inputs, and what error would change the answer?

A surrogate is a replacement constructed to reproduce selected responses of a source model. It may be an interpolation table, a constrained formula, a reduced representation of outputs, a learned function, or a correction to a cheaper model. The practical gain is a less costly response with a usable account of where substitution holds and where to return to the source.

This is a method of mathematical modeling. It applies to computational, physical, biological and organizational models; a differential equation or a neural network is one possible case. It governs the construction and use of a replacement, while the subject practice supplies the meaning and grounds of the source model. Agreement with that model and agreement with the world remain different questions.

Preparation requires functions, interpolation and an error comparison suited to the receiving calculation. The derivative branch uses elementary differentiation; the output-basis branch uses orthogonal projection and singular value decomposition. The stochastic branch requires expectation, variance and the meaning of interval coverage. For a learned replacement, CMP.7 supplies the learning rule and its further-use conditions, with CMP.6 supplying iterative computation when needed. A specialist can construct a harder approximation, but must return an evaluable replacement and the conditions under which its response can be substituted.

For a few inexpensive queries, evaluate the source or reuse already computed answers. If a prescribed numerical approximation already supplies the response and sufficient error control, CMP.8 may finish the work. Surrogate construction becomes useful when selecting the responses, cases and retained structure changes what can be obtained at affordable cost.