# A learned field still obeys an evidence boundary Canonical: https://semiaifoundry.com/learn/atlas/scientific-surrogate-validation/ Revision: 2026-10-07 Rights: https://semiaifoundry.com/ai-atlas/licenses.html Models become evaluated applications · Atlas v1.3.0 # A learned field still obeys an evidence boundary Train on a bounded parameter family; compare prediction with a numerical reference, boundary conditions and conservation residuals. [Explore in Atlas](https://semiaifoundry.com/ai-atlas/#scientific-surrogate-validation)Logical state, task evidence and declared units; no physical assembly scale Learning objectiveCompare a predicted field with matching boundary conditions, local error and an explicit discrete residual. ## The mechanism A scientific surrogate learns an approximation to a family of physical input-output relationships. Its prediction should be compared with an identified reference problem, including geometry, boundary conditions, units and parameter range. A small average field error can hide an important local error, and an output that resembles a physical field need not satisfy the governing equation. Neural operators such as the Fourier Neural Operator provide one approach to learning mappings between function spaces. The worked heat example below does not train or run an FNO; it uses a three-point field to separate value error from a discrete equation residual. Residual checks themselves have a resolution and model boundary and cannot establish every physical property or out-of-distribution capability. 1. The reference problem fixes geometry, coefficients, forcing and boundary conditions before any learned output is assessed. 2. Training examples and evaluation parameters remain distinct, with the valid parameter family stated. 3. Predicted fields are compared both with reference values and with relevant equation or conservation checks. 4. Unseen boundaries or parameter ranges are evaluated explicitly rather than inferred from visual smoothness. ## Science and assumptions ### A learned field still obeys an evidence boundary · explicit relationship $$ r_i=\frac{T_{i-1}-2T_i+T_{i+1}}{(\Delta x)^2} $$ The centered second-difference residual estimates the one-dimensional temperature curvature at an interior point. Symbols and units rᵢ Discrete residual for the source-free steady curvature condition K/m² Tᵢ₋₁, Tᵢ, Tᵢ₊₁ Neighboring grid temperatures K Δx Positive uniform spatial spacing m Consider a one-dimensional, steady, source-free conductor with three equally spaced points at x=\[0,0.5,1\] m and boundary temperatures 293 and 393 K in this mathematical example. The linear reference is \[293,343,393\] K. A hypothetical surrogate gives \[293,353,393\] K. Mean absolute error across the three points is 10/3 K, but the interior error is 10 K. Its centered second-difference residual is (293−2×353+393)/(0.5²)=−80 K/m², whereas the reference residual is zero. This is a discrete steady-equation check, not measured heat flow or a proof of full continuum accuracy. Assumptions Constant-conductivity, one-dimensional, steady, source-free mathematical example with fixed boundaries. The discrete residual is not a complete PDE validation or a field measurement. ## Engineering example Consider a one-dimensional, steady, source-free conductor with three equally spaced points at x=\[0,0.5,1\] m and boundary temperatures 293 and 393 K in this mathematical example. The linear reference is \[293,343,393\] K. A hypothetical surrogate gives \[293,353,393\] K. Mean absolute error across the three points is 10/3 K, but the interior error is 10 K. Its centered second-difference residual is (293−2×353+393)/(0.5²)=−80 K/m², whereas the reference residual is zero. This is a discrete steady-equation check, not measured heat flow or a proof of full continuum accuracy. ## Distinctions and tradeoffs A smooth or low-average-error field is not automatically a valid physical solution. Surrogates can amortize repeated approximations but need evidence across their claimed parameter family and may fail under new boundaries. AI for science is meaningful when learned predictions are tested against the actual physical problem and its limits. ## Specifications and interfaces Mechanism boundary Where does a surrogate cease to approximate the physical problem? Constant-conductivity, one-dimensional, steady, source-free mathematical example with fixed boundaries. The discrete residual is not a complete PDE validation or a field measurement. Evidence type Authored worked example Worked explanation with a bounded numeric or state example. This lesson does not claim an executing implementation of an external model or platform. ### Interfaces Declared input and state Input The reference problem fixes geometry, coefficients, forcing and boundary conditions before any learned output is assessed. Checked outcome Output Compare a predicted field with matching boundary conditions, local error and an explicit discrete residual. ## Check your understanding ### What is the interior second-difference residual for \[293,353,393\] K with spacing 0.5 m? 1. 0 K/m² 2. −80 K/m² 3. −20 K Answer and reasoning −80 K/m² The numerator is −20 K and division by 0.25 m² gives −80 K/m². ## Evidence and scope This is a worked explanation, not an executing external platform, model or robot. The quantities and fixtures are authored. Source scope is recorded in the automation learning review manifest; no source benchmark or private system behavior is claimed. Constant-conductivity, one-dimensional, steady, source-free mathematical example with fixed boundaries. The discrete residual is not a complete PDE validation or a field measurement. The worked example explains this mechanism. General numerical labs are a reference; they do not execute this lesson’s full mechanism. ## Sources - [Fourier Neural Operator for Parametric Partial Differential Equations](https://arxiv.org/abs/2010.08895v3) [Published lesson data](https://semiaifoundry.com/ai-atlas/lesson-data/scientific-surrogate-validation-4b003bf0f160b9af.json) · [Atlas releases and licensing](https://semiaifoundry.com/ai-atlas/licenses.html) ## Continue exploring ### Foundations - [Evaluation & the AGI frontier](https://semiaifoundry.com/learn/atlas/evaluation/) - [Training data pipeline](https://semiaifoundry.com/learn/atlas/training-data/) ### Related lessons - [Tools, memory & agent loop](https://semiaifoundry.com/learn/atlas/tool-loop/)