Derivative-observation Gaussian process regression
Algorithms AIMech:DerivativeObservationGaussianProcessRegression DRAFT
Aliases: Gaussian process regression with derivative observations
A Gaussian-process regression procedure that jointly conditions a sufficiently differentiable scalar-function prior on function values and partial-derivative values using differentiated kernel covariance blocks and a specified Gaussian observation-error model.
Identity Notes & Concept Scope
Compared all 442 pinned AIO IDs, labels and scoped synonyms and 539 native IDs, names and aliases against the proposed name, ID, descriptive variants and neighboring gradient-enhanced terminology: no normalized identity match. Ignored- and hidden-inclusive full-text searches in the recovery worktree, older discovery worktree and original Documents checkout returned 23, zero and zero lines. All returned lines were inspected; they contain virtual-sign contextual mentions, a pending derivative-observation cohort or an unrelated recursive-estimation derivative calculation. Read all 34 semantic AIO rows, 66 native name/definition neighbors, five complete annotated AIO CLASS nodes with incident edges and their generated records, and both complete native GP neighbors. Generic GPR does not require derivative observations; the virtual-derivative monotonicity algorithm uses non-Gaussian sign likelihoods and EP. This record specifies Gaussian conditioning on derivative values. Local-linearization summaries are an optional source of those observations, not a second identity established here. Direct gradient-enhanced kriging may overlap; the umbrella also includes indirect Taylor pseudo-observations, so no blanket alias is asserted without its own full-method comparison. The absence assessment is bounded to the inspected snapshot and records.
Claims & Grounded Evidence
5 Claims
Mechanism
#joint-kernel-derivatives
For a sufficiently mean-square differentiable scalar GP f, form a joint Gaussian vector of function values and selected partial derivative values. Its covariance blocks are k(x,x_prime), derivatives with respect to the appropriate first or second kernel argument, and mixed derivatives with respect to both arguments. Differentiate the latent kernel, then add the observation-error covariance. The inspected squared-exponential construction gives cov(f_prime(x),f(x_prime))=-(x-x_prime)*exp(-(x-x_prime)²/2) in the one-dimensional unit-kernel case. Modeling all derivatives through the same scalar function enforces mutual integrability of the latent derivative field; it does not force noisy observed gradients to be matched exactly.
Mechanism
#gaussian-conditioning
With fixed kernel parameters and additive jointly Gaussian observation errors independent of the latent function, combine the value and derivative observations in one Gaussian conditioning calculation. A full error covariance can retain correlations among observations. Posterior means and covariances are available for both values and derivatives. The paper fits kernel hyperparameters by joint marginal likelihood; conditional exactness under the specified Gaussian model does not integrate hyperparameter uncertainty or guarantee a globally optimal fit.
Mechanism
#local-linearization-summaries
An optional system-identification construction fits a local linear model to perturbation data around an operating point. Its intercept supplies a function-value observation and its D slopes supply D partial-derivative observations, with the estimated full (D+1)-dimensional coefficient covariance added as an error block. Treating different blocks as independent requires the corresponding error assumption. These summaries can reduce the dense GP system size when many locally linear observations are replaced by fewer coefficients; they are not generally a lossless representation of arbitrary nonlinear data.
Mechanism
#mean-and-uncertainty
Derivative observations can change both the posterior mean and its uncertainty. For a zero-mean unit squared-exponential GP conditioned on exact f(0)=1 and f_prime(0)=d, the mean at x is exp(-x²/2)*(1+x*d) and latent variance is 1-exp(-x²)*(1+x²). The zero-derivative symmetric example leaves the value-only mean unchanged, but d=1 changes it. At x=1 the variance is 1-2/e, about 0.2642. These are fixed-parameter curator conditioning calculations, not a reproduction of the paper’s fitted examples.
Limitation
#finite-difference-and-dependence
Two exact function values at distinct locations are not literally the same observations as an exact derivative and a function value. With the unit kernel, f(0) and f(1) fix the variance at x=1 to zero, whereas f(0) and f_prime(0) leave variance 1-2/e. The source’s exact-equivalence wording therefore requires a derivative-limit qualification. Estimated slopes and reused raw data can also share errors: treating them as independent evidence can misstate uncertainty. A curator scalar Gaussian example gives posterior variance 1/2 from one noisy observation, but 1/3 when a copy is incorrectly counted as independent.
⚠️ Curated Limitations & Negative Results
- This record scopes direct Gaussian conditioning on derivative values, including noisy estimates, with function values. Virtual derivative signs with EP, finite-difference pseudo-observation augmentation, a separately fitted GP for every gradient component, and particular nonlinear controllers are separate constructions. No universal gradient-enhanced-kriging alias is assigned.
- The latent kernel must support the required mean-square derivatives. Gaussian conditional formulas assume a valid covariance and specified error model; nonsmooth kernels, correlated errors treated as independent, ill-conditioning and duplicate constraints need appropriate handling. Smooth latent integrability is not a proof of dynamical stability, controllability or globally accurate identification.
- Local-linearization compression needs a justified local approximation and error covariance. Equations 7–9 give a particular regularized fit and covariance estimate; they do not establish exact uncertainty for every prior scaling or every regularized estimator. Correlation within coefficient blocks and with any retained original data matters. Local equilibrium models alone do not identify general off-equilibrium dynamics.
- At fixed Gaussian hyperparameters, adding observations cannot increase conditional covariance, but refitting hyperparameters changes the comparison. The source’s near-equivalence of 150 raw observations and a summary is a particular illustration, not a sufficiency theorem or universal accuracy/computational gain. The general cubic dense-system cost uses the total value and derivative observation count.
- The source’s two-nearby-values equivalence and zero-derivative mean wording are qualified by explicit curator calculations. The nonlinear demonstration’s output-time indexing and noise notation need resolution before any exact numerical reproduction. No benchmark, dynamical simulation, fitted GP or author implementation was executed; the record remains DRAFT after author self-review.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:RegressionAnalysis
Regression Analysis
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related_match | The procedure estimates a response relationship using function and derivative observations. RegressionAnalysis names the broader task, whereas this record specifies an inference algorithm; the mapping does not assert exact identity or an upstream subclass. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
solak2002
|
E. Solak, R. Murray-Smith, W. E. Leithead, D. J. Leith, C. E. Rasmussen
|
primary paper
full_text
|
ID: https://proceedings.neurips.cc/paper_files/paper/2002/hash/5b8e4fd39d9786228649a8a8bec4e008-Abstract.html
sha256:72c7205d979b…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-12T20:48:44Z
Updated: 2026-10-12T20:48:44Z