MMP.17:4.3 - Construct an evaluable replacement with the needed structure

Start with a representation that can express the required response at the available construction cost. Use MMP.11 to preserve justified relations and to expose what remains adjustable.

For ordered scalar cases ((x_i,r_i)), one complete construction is piecewise linear interpolation:

[ \widehat r(x) =\frac{x_{i+1}-x}{x_{i+1}-x_i}r_i +\frac{x-x_i}{x_{i+1}-x_i}r_{i+1}, \qquad x_i\leq x\leq x_{i+1}. ]

The rule supplies a response between distinct neighboring inputs. It needs no iterative training. Its adequacy between the cases still depends on the response’s variation and the receiving tolerance.

For a field or a large output vector, one can instead construct

[ \widehat y(x)=y_0+\sum_{j=1}^{k} a_j(x)\phi_j. ]

Here (y_0) and the retained output shapes (\phi_j) come from known structure or computed cases. For example, take (y_0) as the mean case vector, stack the centered case vectors as columns, retain selected left singular vectors of that matrix, and project each centered case onto those orthonormal vectors. Then interpolate or learn the coefficient functions (a_j(x)) from the inputs and the computed coefficients. A small reconstruction error over sampled fields can still discard a localized feature that controls the receiver’s maximum or threshold. If only (Q(y)) is needed, compare approximating that response directly with reconstructing the full field.

When a cheap model (L) already follows much of the response, construct a correction from paired cases:

[ d_i=r(x_i)-L(x_i),\qquad \widehat r(x)=L(x)+\widehat d(x). ]

Pair the same inputs and corresponding outputs. This construction still evaluates (L) at each new input. It is useful when the discrepancy is easier to approximate than the whole response; if the cheap model misses the consequential regime, adding many cheap cases may help little.

For a learned function, supply CMP.7 with the target, construction cases, function family and loss that reflects the required response. Obtain its effective fitting procedure through CMP.6 or another suitable computation. Writing an objective without a way to obtain and evaluate its candidate leaves the replacement unfinished. Optimization progress, fit on construction cases and accuracy at further inputs are separate results.

Preserve a justified relation by construction when possible. If two delivered quantities must sum to an input (d), construct one and define the other as (d-\widehat q_1), while also enforcing any required nonnegativity or capacity limits. A small penalty for violating a relation permits violations; it does not implement the relation as an identity.