Scientific AI and Numerical Methods Reference

Purpose

This document is the canonical repository reference for combining:

• numerical methods from Numerical Recipes;
• neural representation and architecture from Bishop and Bishop;
• probabilistic inference and decision theory from Murphy;
• domain-specific forward models and experimental evidence.

The governing pattern is:

numerics + neural representation + probabilistic inference = trustworthy scientific prediction

Code should reference this document when an implementation choice depends on these methodological principles.

Quick start: apply the method

For a new scientific prediction task, work through these steps in order:

1. define the observable, units, admissible inputs, and decision that the prediction must support;
2. implement the simplest defensible numerical or physical baseline;
3. add the learned representation only where it provides measurable value over that baseline;
4. expose uncertainty, applicability-domain status, provenance, assumptions, and limitations with every prediction;
5. validate on held-out or prospective evidence and block unsupported extrapolation;
6. record the relevant method anchor in code and tests;
7. propose the next measurement or simulation when the evidence is insufficient.

Start with the prediction contract. Then use the relevant domain map for SAXS, SPR, or cryo-EM and finish with the validation contract.

Literature and data graph

The calibration literature and data graph turns the references, datasets, and reusable ideas around Guo et al. (2017) into a searchable evidence index.

Its YAML records keep four entities distinct:

• papers and their role in the method lineage;
• datasets and the split reported by the paper;
• extracted ideas with source locations and operational consequences;
• typed relations between evidence records.

Use the machine-readable JSON index when code, agents, or analytical tools need to scan this evidence. Treat an extracted idea as a traceable interpretation, not as a replacement for reading the primary source.

How code should reference this contract

Prefer a short comment next to the relevant interface, model, validation gate, or numerical operator:

Method: CALYR-METHOD-NR-AI-001
See: docs/scientific-ai-numerics-reference.md#prediction-contract

Use a more specific anchor where possible:

• #numerical-baseline-before-learned-model
• #prediction-contract
• #applicability-domain
• #saxs-method-map
• #spr-method-map
• #cryo-em-method-map
• #numerical-recipes-code-policy
• #validation-contract

Do not paste long explanations into source files. Keep the reasoning here and make code comments point to the stable section.

Division of responsibility

No one layer may silently replace another. A neural network does not replace the physical definition of an observable. A numerical fit does not by itself establish identifiability. A posterior does not become a decision without an explicit loss, constraint, or utility rule.

Numerical baseline before learned model

Every learned scientific model should be compared with the simplest defensible numerical baseline.

The baseline should define:

1. inputs, outputs, units, and admissible ranges;
2. forward model or observation operator;
3. objective or likelihood;
4. optimization or inference method;
5. numerical tolerances;
6. uncertainty estimate;
7. known non-identifiability;
8. failure behavior outside the calibrated domain.

Typical reference operations include SVD, nonlinear least squares, FFT, convolution, correlation, ODE integration, regularization, Gaussian-process regression, and MCMC.

Prediction contract

Scientific prediction is a structured result, not a bare value. Every public prediction interface should expose, directly or through linked metadata:

prediction:
  target: string
  value: scalar | array | field | distribution
  unit: string
  interval:
    lower: number
    upper: number
    level: number
applicability:
  status: in_domain | boundary | out_of_domain
  score: number
  reasons: [string]
model:
  id: string
  version: string
  data_snapshot: string
evidence:
  references: [string]
  assumptions: [string]
  limitations: [string]
decision:
  status: usable | review_required | blocked
  next_action: string

The exact serialization may vary, but the semantics must remain available and testable.

Applicability domain

A model must distinguish interpolation within validated conditions from unsupported extrapolation.

Applicability checks should consider:

• distance from training and validation observations;
• physical parameter limits;
• experimental protocol and instrument conditions;
• structural or mechanistic class;
• preprocessing compatibility;
• uncertainty and ensemble disagreement;
• missing inputs and violated constraints.

Out-of-domain results must not be rendered as ordinary predictions. They should be marked review_required or blocked and should propose a simulation, measurement, or expert review.

SAXS method map

Numerical layer

• quadrature and multidimensional integration;
• Bessel functions and spherical harmonics;
• Fourier transforms;
• convolution for resolution smearing;
• SVD and regularization for inverse problems;
• nonlinear fitting and MCMC.

Neural layer

• curve encoders or transformers for I(q);
• continuous latent variables for structural ensembles;
• graph or geometric representations for molecular structure;
• forward surrogates for structure -> I(q).

Probabilistic layer

• posterior inference over competing structures or ensembles;
• calibrated predictive distributions;
• model comparison and posterior predictive checks;
• active selection of q-range, contrast, concentration, or next simulation.

The implementation must not claim unique structural recovery where the SAXS evidence is non-identifying.

SPR method map

Numerical layer

• nonlinear least squares;
• robust estimation;
• ODE integration for kinetic and transport models;
• conservative filtering with distortion tests;
• sensitivity and identifiability analysis.

Neural layer

• sensorgram sequence encoders or transformers;
• structured prediction of ka, kd, KD, Rmax, and full response curves;
• neural surrogates for repeated kinetic simulation.

Probabilistic layer

• posterior inference over parameters and model class;
• latent-state or state-space models;
• sequential updating across cycles and concentrations;
• next-assay selection by expected information or decision value.

The implementation must keep kinetic, mass-transport, rebinding, heterogeneity, drift, and nuisance explanations distinguishable.

Cryo-EM method map

Numerical layer

• multidimensional FFT;
• convolution and correlation;
• orientation and rotation geometry;
• SVD/eigensystems for low-rank structure;
• regularized inverse reconstruction.

Neural layer

• CNNs and transformers for particles and maps;
• autoencoders for continuous heterogeneity;
• GNNs for molecular structure;
• conditional generative models for reconstruction or refinement.

Probabilistic layer

• latent orientation and conformation inference;
• posterior predictive image or map checks;
• uncertainty over structure, resolution, and heterogeneity;
• distribution-shift detection across acquisition and preprocessing conditions.

The implementation must separate resolution, noise, heterogeneity, orientation uncertainty, preprocessing effects, and model bias.

Numerical Recipes code policy

The current Third Edition code has separate licensing conditions. Do not copy it into a CALYR repository without an explicit license review.

The official legacy routines are useful as algorithmic references but are obsolete and may use deprecated conventions. Do not treat them as production-ready dependencies.

Preferred production targets include maintained implementations in:

• NumPy and SciPy;
• scikit-learn;
• PyTorch;
• JAX;
• PyMC, NumPyro, BlackJAX, or Stan;
• domain-specific validated libraries.

Useful routine mappings from the official routine index include:

When an implementation is inspired by a Numerical Recipes algorithm, record the edition, section, routine name, source URL, license status, maintained replacement, and verification tests.

Validation contract

Every surrogate or probabilistic prediction model should be evaluated on:

• held-out and prospective prediction;
• comparison with a transparent baseline;
• calibration and interval coverage;
• applicability-domain detection;
• sensitivity to preprocessing and nuisance variables;
• physical constraint violations;
• reproducibility across data splits and random seeds;
• posterior predictive checks;
• failure modes under distribution shift;
• value of the proposed next experiment or simulation.

Training fit alone is not validation.

Recommended repository pattern

docs/
  scientific-ai-numerics-reference.md
references/
  numerical-recipes/
    README.md               # citation, routine mapping, license notes
tests/
  reference_cases/          # independently authored numerical cases
src/
  numerics/                 # maintained numerical implementations
  surrogates/               # neural representations
  inference/                # probabilistic inference
  validation/               # calibration and applicability gates

This contract is implementation-language independent and may be referenced from Python, C++, JavaScript, YAML, notebooks, tests, and design documents.