Validation

Lattent's analysis engine is a small, readable JavaScript library. The same reference datasets are fitted by Lattent and by SciPy, the standard scientific Python library, and the results must agree before any release. The test suites (tests/run-engines.js with tests/check_scipy.py for the calculators, and tests/run-studio.js with tests/check_studio.py for Studio) simulate data with known parameters and noise, fits it in both, and compare estimates, standard errors, derived quantities and p-values. A technical note further down benchmarks the fitting engine against certified reference results.

Check	Reference	Largest relative difference
Dose–response, 4PL inhibitor (30 points)	scipy.optimize.curve_fit	pass parameters 1×10⁻⁸, SE 3×10⁻⁸
Dose–response, 4PL agonist	curve_fit	pass parameters < 10⁻⁶, SE < 10⁻⁵
Dose–response, 5PL asymmetric	curve_fit	pass parameters < 10⁻⁶, SE < 10⁻³
EC80 / IC80 from the fitted curve	closed-form inverse of the 4PL	pass < 10⁻⁶
Standard curve, 4PL unweighted	curve_fit	pass parameters < 10⁻⁶, SE < 10⁻³
Standard curve, 4PL with 1/Y² weighting	curve_fit with sigma = Y	pass parameters < 10⁻⁶, SE < 10⁻³
Sample interpolation	analytic 4PL inverse	pass < 10⁻⁶
Linear standard curve	scipy.stats.linregress	pass parameters < 10⁻⁸, SE < 10⁻⁵
Michaelis–Menten	curve_fit	pass parameters < 10⁻⁴, SE < 10⁻³
Substrate inhibition, Hill	curve_fit	pass parameters < 10⁻⁴
Competitive, non-competitive, uncompetitive, mixed (global)	curve_fit on stacked data	pass parameters < 10⁻⁴, SE < 10⁻³
Mechanism selection by AICc	known simulated mechanism	pass competitive correctly preferred
Student t CDF and quantiles	scipy.stats.t	pass < 10⁻⁷
Welch's t-test	scipy.stats.ttest_ind(equal_var=False)	pass < 10⁻⁶
qPCR fold change (2^−ΔΔCt)	hand calculation	pass < 10⁻⁹
qPCR p-value	ttest_ind on −ΔΔCt	pass < 10⁻⁶

Lattent Studio engine

Check	Reference	Largest relative difference
Normality (D'Agostino–Pearson K²)	scipy.stats.normaltest	pass < 10⁻⁶
Student, paired t tests	ttest_ind, ttest_rel	pass < 10⁻⁷
Mann–Whitney, exact and asymptotic	mannwhitneyu	pass < 10⁻⁷
One-way ANOVA, Kruskal–Wallis, Brown–Forsythe	f_oneway, kruskal, levene	pass < 10⁻⁷
Welch's ANOVA	textbook formula in NumPy	pass < 10⁻⁷
Tukey HSD p-values and intervals	scipy.stats.tukey_hsd	pass < 2×10⁻³ (numerical integration)
Studentized range distribution	scipy.stats.studentized_range	pass < 10⁻³
Pearson and Spearman correlation	pearsonr, spearmanr	pass < 10⁻⁷
Multiple linear regression	numpy.linalg.lstsq	pass < 10⁻⁸
PCA eigenvalues	numpy.linalg.eigvalsh	pass < 10⁻⁸
Arrhenius, Langmuir, KWW, Carreau fits	scipy.optimize.curve_fit	pass parameters < 10⁻⁵, SE < 2×10⁻³
User-defined equation parser	curve_fit on the same expression	pass < 10⁻⁵
Multi-peak fit (simulated XRD)	known peak positions and widths	pass centres < 0.2%, widths < 5%
d-spacing and Scherrer size	Bragg and Scherrer equations	pass < 10⁻⁹
Savitzky–Golay smoothing	scipy.signal.savgol_filter	pass < 10⁻⁹
Gaussian-process mean, SD and leave-one-out	closed form in NumPy	pass < 10⁻⁵
Tauc band gap (simulated, 3.20 eV)	known value	pass within 0.05 eV
Tensile modulus (simulated, 70 GPa)	known value	pass within 2%

Run them yourself: node tests/run-studio.js out.json && python tests/check_studio.py out.json.

Technical note: nonlinear fitting on the NIST reference problems

Measured 16 October 2026 on an Apple M1 · Node 24.4 (V8, the engine in Chrome and Edge) · Python 3.12.9 with SciPy 1.18.1

Agreeing with SciPy on well-behaved data shows that two implementations match; it does not show that either is right on hard problems. For that, the US National Institute of Standards and Technology publishes the Statistical Reference Datasets (StRD): nonlinear regression problems with certified parameter values computed to 11 significant digits, each with two prescribed starting points, the first further from the solution. Lattent's fitting engine (the code behind every Studio fit and user-defined equation) and scipy.optimize.curve_fit were run on all 26 single-predictor problems from both starting points: 52 fits.

51 / 52Lattent fits with at least 4 correct significant digits in every parameter

45 / 52SciPy curve_fit, default settings

8.6 vs 6.4median correct digits, worst parameter of each fit

17 vs 0.6 msmedian time per fit; SciPy is faster

NIST difficulty	Fits	Fits with ≥ 4 digits		Median digits
NIST difficulty	Fits	Lattent	SciPy	Lattent	SciPy
Lower	16	16	16	10.1	7.8
Average	20	20	17	9.0	6.4
Higher	16	15	12	7.7	5.1
Standard errors (all fits)	52	47	43	8.4	5.1

Reading the numbers. Accuracy is the log relative error, −log₁₀(|estimate − certified| / |certified|), roughly the number of correct significant digits, taken for the worst parameter of each fit and capped at 11. Four digits is the usual pass mark; anything above 6 is more precision than any laboratory measurement supports.

Where Lattent does better. It converges from the harder first start on MGH17, BoxBOD and Eckerle4, and to five digits on ENSO, where SciPy's default tolerances stop at three. Two design choices account for this. First, Levenberg–Marquardt with Moré's diagonal scaling and “delayed gratification” damping (Transtrum and Sethna, 2012). Second, a few short scouting runs from perturbed starting points, the best of which is refined to convergence. Its convergence tolerance is also tighter than SciPy's default, which explains the extra digits on easy problems.
Where it does worse. MGH10 from the first start (b₂ = 400,000 against a solution near 6,181) does not converge; SciPy solves it. The failure is not silent: Lattent flags all three parameters as poorly determined and the residuals as systematic (runs test p = 0.003), and the second start succeeds.
Speed. SciPy runs compiled MINPACK Fortran and, with its default tolerances, stops earlier: it is about 20 times faster at the median and up to several hundred times faster on the largest problems. Lattent evaluates the equation in JavaScript with numerical derivatives and runs the extra scouting starts: 39 of 52 fits take under 0.1 s, and the slowest (Gauss3, 250 points and 8 parameters) takes 0.77 s. On a laptop the difference is not noticeable for interactive work.
Not compared here. SciPy run with tighter tolerances or several starts would gain digits too; this compares each tool as a user meets it. Rietveld refinement programs (GSAS-II, FullProf) solve a different problem, refining a full crystal structure against a whole pattern, and are not benchmarked: Lattent's XRD tools fit peaks and match phases, they do not do Rietveld refinement.

All 52 fits

Problem	Difficulty	Start	Lattent digits	SciPy digits	Lattent ms	SciPy ms
Misra1a	Lower	1	11.0	8.6	3.3	0.3
Misra1a	Lower	2	10.9	8.4	2.4	0.2
Chwirut2	Lower	1	8.6	6.2	6.1	0.2
Chwirut2	Lower	2	8.5	5.9	5.5	0.2
Chwirut1	Lower	1	7.8	5.8	22	0.3
Chwirut1	Lower	2	8.6	5.9	16	0.2
Lanczos3	Lower	1	6.8	4.6	54	5.1
Lanczos3	Lower	2	7.2	4.8	51	0.6
Gauss1	Lower	1	10.6	6.9	720	0.8
Gauss1	Lower	2	10.6	6.8	730	0.8
Gauss2	Lower	1	9.9	8.5	510	1.0
Gauss2	Lower	2	9.9	8.1	467	0.9
DanWood	Lower	1	11.0	7.9	1.4	0.1
DanWood	Lower	2	11.0	8.7	1.4	0.1
Misra1b	Lower	1	10.2	7.8	3.7	0.5
Misra1b	Lower	2	10.2	7.9	2.6	0.2
Kirby2	Average	1	8.2	6.4	34	0.5
Kirby2	Average	2	8.3	5.8	32	0.4
Hahn1	Average	1	5.1	6.4	220	2.0
Hahn1	Average	2	5.0	5.1	226	1.9
MGH17	Average	1	9.0	failed	130	0.2
MGH17	Average	2	9.5	6.4	10	0.8
Lanczos1	Average	1	10.6	10.6	53	4.7
Lanczos1	Average	2	10.6	10.6	50	0.6
Lanczos2	Average	1	6.8	6.6	52	5.1
Lanczos2	Average	2	7.7	6.4	52	0.6
Gauss3	Average	1	10.0	9.2	765	1.1
Gauss3	Average	2	10.1	7.7	695	1.2
Misra1c	Average	1	10.9	8.2	2.6	0.2
Misra1c	Average	2	10.8	7.9	2.4	0.1
Misra1d	Average	1	11.0	8.7	2.8	0.2
Misra1d	Average	2	11.0	8.4	2.3	0.1
Roszman1	Average	1	8.3	5.5	3.5	0.2
Roszman1	Average	2	8.3	5.6	3.4	0.2
ENSO	Average	1	5.5	3.2	436	6.3
ENSO	Average	2	5.5	3.1	593	5.5
MGH09	Higher	1	6.3	4.0	17	12
MGH09	Higher	2	6.4	4.0	7.0	0.6
Thurber	Higher	1	7.0	4.5	65	3.3
Thurber	Higher	2	6.9	4.5	78	2.1
BoxBOD	Higher	1	7.7	failed	3.5	0.1
BoxBOD	Higher	2	7.8	5.1	1.5	0.2
Rat42	Higher	1	9.9	6.8	2.0	0.2
Rat42	Higher	2	9.7	7.1	1.8	0.2
MGH10	Higher	1	failed	6.7	60	4.6
MGH10	Higher	2	8.8	6.8	13	2.2
Eckerle4	Higher	1	9.6	7.2	7.2	0.4
Eckerle4	Higher	2	9.6	7.2	3.5	0.2
Rat43	Higher	1	7.8	5.1	6.0	0.7
Rat43	Higher	2	7.4	4.9	3.8	0.3
Bennett5	Higher	1	7.3	5.4	168	22
Bennett5	Higher	2	7.5	4.0	198	5.9

Reproduce it: node tests/bench/nist.js and python tests/bench/nist_scipy.py; the data and certified values are in tests/bench/nist.json, converted from NIST's files by tests/bench/extract_nist.py, and the raw results above are in tests/bench/results.json. Continuous integration fails if fewer than 50 of the 52 fits reach four digits. Nelson, the one StRD problem with two predictors, is omitted.

Cite Lattent

Lattent's specialist models

Facet's built-in suggestions and the shape-guided auto-fit use two small neural networks trained by Lattent, not by a third party. Both run entirely in your browser.

Shape model. It recognises 13 families of behaviour from XY data. Training used 67,600 simulated datasets with realistic ranges, spacings, replicates, outliers and noise. On 7,800 held-out datasets it is 98.4% correct first time and 99.8% within its top two. It also classifies all 25 realistic lab examples in the test suite correctly, such as a five-point Arrhenius set, XPS with a step background and a dilution-series calibration line.
Scatter model. It judges whether measurement error grows with the signal, so that 1/Y² weighting is advised. Held-out accuracy is 72% (AUC 0.82). Without replicates this is a hard problem, so Lattent acts on it only when the model is at least 80% confident and the data span a wide range.

Shape family	Held-out accuracy
Straight line	99.3%
Polynomial curvature	95.5%
Decay	99.3%
Rise to a plateau	97.0%
Sigmoidal	97.2%
Exponential growth	98.8%
Power law	96.5%
Logarithmic	99.0%
Thermally activated	100.0%
Inverse (1/X)	98.0%
Spectrum with peaks	98.7%
Absorption edge	99.8%
Stress–strain	100.0%

The models suggest; they never compute results. Every number still comes from the validated fitting engine above. The browser inference is checked against scikit-learn on every release (largest difference below 10⁻⁵). Retrain with node tests/ml/simulate.js and python tests/ml/train.py.

Equation discovery, AutoML and uncertainty

These checks use synthetic data with known answers, so success is unambiguous (tests/ml/check_learn.js).

Check	Data	Result
Equation discovery: six known laws	quadratic, exponential decay, power law (5% relative noise), Arrhenius, Langmuir and Hall–Petch, each with noise	All six recovered with the right form; for example exponent 1.501 (true 1.5), 4029 K (true 4000), K = 3.24 (true 3.33)
Equation discovery on an example	glass transition against molar mass	Rediscovers Flory–Fox: T_g = 100.6 − 1.11×10⁵/M_n (true 100.5 − 1.1×10⁵/M_n)
AutoML: nonlinear benchmark	Friedman #1, 160 samples, 5 informative and 5 pure-noise inputs	Gaussian process selected, cross-validated R² 0.93; the five informative inputs all rank above the five noise inputs
AutoML: linear problem	40 samples, y = 2x₁ − x₂ + noise	A linear model selected, cross-validated R² 0.99
Conformal prediction intervals	90% intervals from AutoML, tested on 400 new samples	89.3% coverage
Property-network intervals	9,511 compositions never seen in training	90% half-widths: 0.17 eV/atom (formation energy); 0.41, 1.15 and 1.29 eV (band gap, for predicted gaps below 0.3, 0.3–2 and above 2 eV)

Equation discovery. Sparse symbolic regression: up to three terms from a library of powers, reciprocals, square roots, logarithms, and exponential and saturation terms on grids of scale parameters, combined linearly for y and (power-type terms only) for ln(y). Candidates are ranked by AICc, on relative errors when y spans more than a factor of 20. The best structures are refined by Levenberg–Marquardt. A recognised law is preferred unless a generic form beats it by more than 14 AICc units, a margin that corrects for the thousands of generic forms searched.
AutoML. Linear and ridge regression (penalty by exact leave-one-out), distance-weighted k-nearest neighbours, a 150-tree random forest, gradient-boosted trees and a Gaussian process with one length-scale per input. Leave-one-out cross-validation for up to 20 samples, five-fold above; constant and duplicate inputs are removed first. Importance is the rise in held-out error when an input is permuted. Intervals are split-conformal from held-out residuals.

Discovery and generative models

Discover combines a live database, a powder-diffraction simulator and three neural networks that Lattent trained itself on Materials Project data (via Matbench, CC BY 4.0). The networks run in your browser, and the browser code reproduces PyTorch to within 2×10⁻⁶ (float32 property networks) and 1.2×10⁻³ (float16 generator).

Check	Data	Result
Formation energy of the most stable polymorph, from composition	9,511 test compositions never seen in training (85,607 for training)	MAE 0.092 eV/atom, R² 0.945 (predicting the median gives 0.98 eV/atom)
DFT (PBE) band gap, from composition	10,573 test entries, split by composition (95,540 for training)	MAE 0.32 eV, R² 0.84 (median predictor 1.19 eV)
Generator: unconditional samples	2,526 formulas	97% distinct, 71% absent from the training set; ternaries are over-represented (75% against 52% in the data)
Generator: kesterite family, Cu–Zn–Sn–S	800 samples	Cu₂ZnSnS₄ is the third most frequent formula (15% of samples) and in the top ten after screening
Generator: olivine cathode, Li–Fe–P–O	300 samples	LiFePO₄ is proposed
Generator: band-gap guidance	400 samples per target	mean predicted gap of the top candidates 0.93 eV for a 0.5 eV target, 3.73 eV for 3.5 eV
Powder XRD, fcc Au (a = 4.078 Å, Cu Kα₁)	reference pattern of gold	38.19°, 44.39°, 64.59°, 77.58°, 81.74° with the fcc extinctions; I(200)/I(111) = 48%
Symmetrisation of distorted DFT cells	strained one-atom fcc cell; distorted hcp cell	24 split reflections become the 5 fcc reflections; hcp cell restored to a = b, γ = 120° at the same volume
Phase identification, gold nanoparticles	5 fitted peaks against all Au structures in Alexandria (PBE)	fcc Au, Fm3̅m, score 0.98 (next phase 0.71); lattice 2.2% below PBE, as expected
Phase identification, ZnO	8 experimental peaks against all Zn–O structures	wurtzite, P6₃mc, ranked first (score 0.91)

Database. Searches query the Alexandria database (Schmidt, Marques and co-workers; 5.8 million PBE and 0.4 million PBEsol structures, CC BY 4.0) directly from your browser through the OPTIMADE API. Formation energies, hull distances and band gaps are Alexandria’s DFT values.
XRD simulation. Reflections are generated in the conventional cell after the stored cell is symmetrised to its reported crystal system. Intensities use |F|² with approximate atomic scattering factors, the Lorentz–polarisation factor and an isotropic Debye–Waller factor (B = 0.5 Å²), so positions are exact for the DFT cell and intensities are a guide.
Phase matching. For each candidate a uniform lattice scale between −4% and +4% is refined, because DFT lattices differ from experiment by 1–2%. The score is √(reference intensity covered × measured intensity explained).
Generator. Masked discrete diffusion over five (element, count) slots with confidence-ordered unmasking in ten steps and classifier-free guidance (w = 1.5) on band gap and number of elements; the chemical system is enforced exactly. Candidates must be charge-balanced with common oxidation states and have a negative predicted formation energy.
Limits. Generation is at the level of composition, not crystal structure, and a negative formation energy does not establish stability against competing phases. PBE band gaps are typically 30–50% below experiment. Treat every candidate as a lead for a calculation or a synthesis.

Run the checks yourself: node tests/gen/check.js tests/gen/props_parity.json and node tests/gen/check_xtal.js (the second queries Alexandria; add --offline to skip it). Retrain with tests/gen/train_props.py and tests/gen/train_mdiff.py.

What is computed and how

Fitting: Levenberg–Marquardt nonlinear least squares with central-difference Jacobians and several starting points. Standard errors come from s²(JᵀWJ)⁻¹, as in SciPy and GraphPad Prism.
Confidence intervals: t-distributed with n − k degrees of freedom. IC50 and EC50 intervals are computed on log concentration; derived quantities (ECx, interpolated concentrations, k_cat) use the delta method.
Model comparison: extra-sum-of-squares F test for nested dose–response models; corrected Akaike criterion and Akaike weights for kinetic models.
qPCR: efficiency-corrected relative quantities normalised to the geometric mean of the reference genes (Hellemans et al., Genome Biology 2007), statistics on log₂ scale, Holm correction.

Limits

Lattent is research software. It is not validated for GLP, 21 CFR Part 11 or clinical diagnostic use. Asymptotic confidence intervals assume roughly normal residuals and a well-determined fit; Lattent warns when a parameter is poorly constrained, but judgement is still yours.