1. What machine learning is (the short version)

• Machine learning means fitting a function to data. Instead of writing a formula by hand that maps inputs (composition, an image, a spectrum) to an output (a property value, a defect or no defect), you adjust parameters to the data.
• A neural network is linear transformations and nonlinear functions stacked in layers. Every parameter is moved a little in the direction that lowers the loss (the gap between prediction and answer). The direction for each parameter is found efficiently by backpropagation Sourced [Ref. 1].
• A good fit on the training data does not guarantee a good fit on new samples. What tends to break an evaluation is how the data are split (samples from the same family straddling training and evaluation: "leakage"), the choice of metric, and extrapolating beyond the range of the training data.

2. Three types of learning — supervised, unsupervised, reinforcement

For this introduction, we organise learning goals as reducing error against targets, learning representations without target labels, and learning a policy from actions and rewards. This is our introductory organisation, not a claim that every method belongs in exactly one box. The original backpropagation study [Ref. 1], the individual studies below and official implementation documentation support the specific mechanisms and evaluations discussed here.

The three categories and the materials examples are our introductory organisation, not an experimental result.

View original SVG, values and source annotations

The core of large language model training is self-supervised learning, where the labels are made from the data itself (for example, predicting the next word in a text). We cover it in our explainer on Transformers and large language models. The later stage that adjusts a model with human preferences as a reward (an application of reinforcement learning) is covered in our explainer on post-training and fine-tuning.

3. Model, loss, optimisation — how a neural network "learns"

(1) The model: linear transformations and nonlinear functions stacked up

One layer of a neural network multiplies an input vector x by a weight matrix W, adds a bias b, and passes the result through an activation function φ such as tanh or ReLU (h = φ(Wx + b)). Stacking many such layers gives the network. Because the activation is nonlinear, stacking layers lets the network represent curved functions (this article's commentary). Hornik et al. established the "universal approximation" result: a standard multilayer feedforward network with a hidden layer can approximate functions to any desired accuracy if it has enough hidden units Sourced [Ref. 3]. But this is a statement that a network that approximates the function exists. It does not guarantee that the parameters can be found from data, or that the network will be right on new data (commentary). We did not check the finer conditions of the theorem (the type of activation, the sense of approximation) against the body of the original paper.

(2) The loss: turning "error" into a single number

The loss function L puts the goal of learning into a formula. Regression commonly uses the mean of the squared differences between prediction and answer (mean squared error); classification uses cross-entropy, based on the logarithm of the probability given to the correct class (this article's explanation). The loss is part of the design that decides "what counts as good", and it can be chosen separately from the evaluation metric.

(3) Optimisation: moving the parameters along the gradient

Gradient descent moves the parameters θ in the direction that lowers the loss (the opposite of the gradient), by a step set by the learning rate η (θ ← θ − η∇L). Stochastic gradient descent (SGD) estimates the gradient from a small subset of the data (a mini-batch) rather than all of it. Adam keeps moving averages of the first and second moments of the gradient and adjusts the step per parameter. The original paper gives, as good default settings for the problems it tested, α = 0.001, β₁ = 0.9, β₂ = 0.999 and ε = 10⁻⁸ Sourced [Ref. 2]. These are defaults for the problems in that paper, not values that are best for every dataset (commentary). Hyperparameters such as the learning rate, batch size and the number and width of layers are chosen on validation data, not training data (Section 6).

The training loop is our explanation of calculating a loss and using gradients to update parameters. It is not the complete workflow of a particular implementation.

An example of counting parameters (Our calculation): a fully connected network with 784 inputs, 128 hidden units and 10 outputs (a shape often used for handwritten-digit classification) has 784×128 + 128 + 128×10 + 10 = 101,770 parameters. One forward pass costs about 2 × 101,770 ≈ 200,000 floating-point operations (counting a multiply and an add together as 2, and leaving out activations and similar terms).

4. Backpropagation — following a small numerical example

Backpropagation is the learning procedure reported by Rumelhart, Hinton and Williams in 1986 [Ref. 1]. Starting from a loss that expresses the difference between outputs and targets, it computes backwards how to adjust the weights in each layer. The equations and small numerical example below are our explanation of this procedure.

The mechanism is the chain rule (differentiating composed functions). Starting from "how sensitive the loss is to the output" at the output side, and multiplying back through one layer at a time, gives the gradients for all parameters at a cost of roughly the same order as the forward pass (this article's commentary). The example below is a network with two inputs, two hidden units (tanh) and one output.

• Input x = (1, 2); weights W₁ = [[0.1, 0.2], [0.3, 0.4]] and W₂ = (0.5, 0.6); biases 0; target t = 1.
• The forward pass gives ŷ = 0.7114 and loss L = 0.04166. The backward pass gives, for one component of W₁, ∂L/∂W₁[0,0] = −0.11350.
• Whether this gradient is right can be checked by numerical differentiation: move the weight by a tiny amount (10⁻⁶) and measure the change in loss. Our calculation gives −0.113501, matching the backpropagation value (−0.113501).
• One update at learning rate 0.1 lowers the loss from 0.04166 to 0.02759.

Our calculation. Python (NumPy), double precision. The inputs, weights and target are illustrative values, not real data.

5. Generalisation and overfitting — fitting the training data versus being right on new data

The aim of learning is not to memorise the training data but to be right on data the model has not seen (generalisation). When a model is so complex that it fits even the noise in the data, that is overfitting; when it is too simple to capture even the trend, that is underfitting. The figure below reproduces overfitting on a small synthetic dataset.

• Data: the training set is 15 points of y = sin(2πx) with normal noise of standard deviation 0.2 added. For evaluation we used 2,000 new points made the same way (random seed 0).
• Model: polynomials (the higher the degree, the more freedom), fitted by least squares.
• Result: at degrees 1 and 2 the fit is close to a straight line and misses on both training and test data (underfitting). At degrees 3 to 5 the test error is lowest, about 0.24. As the degree rises further, the training error keeps falling while the test error climbs (0.41 at degree 9).
• Degree 14 (a curve through all 15 points): the training error is almost zero, but the test error jumps to about 8,600 (numerical precision also plays a part).

Our calculation. Synthetic data; not the result for any particular materials dataset. The measurement noise (0.2) is a rough floor that no model can get below (this article's commentary).

(1) Ways to curb overfitting (the general ones)

The usual methods are to add data, make the model smaller, regularise (penalise large weights, as in weight decay), stop early (halt when the validation loss starts to worsen), and use dropout (randomly disabling units during training). Srivastava et al. proposed dropout as a simple way of preventing overfitting in neural networks Sourced [Ref. 4]. The other methods are explained here as introductory terminology.

(2) "More complexity always makes things worse" is not always true

Classical statistics says that a model that is too complex has too much variance and the test error rises (a U-shaped curve). Yet modern neural networks are known to reach high accuracy even when large enough to interpolate the training data exactly. Belkin et al. reconciled the two in a double descent curve: past the point where the model can interpolate the data (the interpolation threshold), adding complexity makes performance better again Sourced [Ref. 6]. Nakkiran et al. showed that double descent occurs not only with model size but also with the number of training epochs, and that there are even regimes where more training data hurts performance Sourced [Ref. 7].

(3) Re-using the same evaluation set has limits too

If the same evaluation set is used for years, researchers may unconsciously choose models and settings to suit that set. Recht et al. rebuilt new test sets for CIFAR-10 and ImageNet by following the original creation procedures closely, and re-measured existing models. Accuracy fell by 3–15% on CIFAR-10 and by 11–14% on ImageNet Sourced [Ref. 8]. At the same time, gains in accuracy on the original test sets translated into larger gains on the new ones. The authors conclude that the drop is due not to adaptation to the evaluation data but to the models' inability to generalise to slightly "harder" images than those in the original test sets [Ref. 8].

The lesson is that a number on an evaluation set is conditioned on how that set was made and used (this article's commentary). Our explainer on benchmarks and how to read evaluations treats this at length for language-model benchmarks.

6. Evaluation basics — how to split the data, and which metric

(1) Training, validation and test sets

Split the data into a training set (to learn the parameters), a validation set (to choose hyperparameters and models) and a test set (used once at the end to report performance). When data are scarce, cross-validation is used: split into training and validation sets several ways and average. The scikit-learn user guide explains that a model which merely repeated the labels of samples it had just seen would score perfectly but be useless on unseen data, and that holding out part of the data as a test set is therefore the usual practice Sourced [Ref. 19].

(2) Choose the metric for the purpose

Metrics depend on the kind of output. Classification uses accuracy, precision, recall, the F1 score and the area under the ROC curve (AUC); regression uses mean absolute error (MAE), root-mean-square error (RMSE) and the coefficient of determination (R²) (this article's commentary). ISO/IEC TS 4213:2022 is a technical specification that sets out methods for measuring the classification performance of machine learning models, systems and algorithms Sourced [Ref. 23] (its text is paid, and we did not check it).

Example: classification for rare defects (Our calculation). Suppose 100 of 10,000 samples (1%) are defective. A do-nothing model that simply answers "good" for everything still has an accuracy of 99%. Now take a model that finds 60 of the 100 defective samples (recall 60%) but wrongly flags 140 of the 9,900 good ones (precision 60 ÷ 200 = 30%). Its accuracy is 98.2%, lower than the do-nothing model's. Its F1 score is 0.40.

Our calculation. The numbers are an illustrative, hypothetical example. With imbalanced data, accuracy alone cannot tell you whether a model is any good. Which metric to use should be decided by the cost of misses (false negatives) and false alarms (false positives) (this article's commentary).

(3) Compare with a simple baseline

A complex model's number alone does not say whether it is good. It means something only when compared with a simple baseline, such as "always return the mean" or "return the value of the nearest training sample". Meredig et al. introduced a simple nearest-neighbour benchmark in evaluating machine learning for materials discovery Sourced [Ref. 11].

7. Cautions for materials engineers — leakage, extrapolation, small datasets

Materials data are often made up of families of samples (a series with the same composition, the same deposition batch, repeated measurements on the same tool), which makes the data split and the evaluation prone to pitfalls (this article's commentary).

(1) Data leakage

Kapoor and Narayanan report that, in the research communities that use machine learning, data leakage was found in 17 fields, affecting at least 294 papers. They sort leakage into eight types: no test set, pre-processing across training and test sets, feature selection across them, duplicated data, illegitimate features, temporal leakage, non-independence between training and test samples, and a test distribution that is biased Sourced [Ref. 10]. The typical case with materials data is the same sample, or samples from the same family, falling in both training and evaluation. This is Kapoor et al.'s "non-independence between train and test samples" (L3.2). As a remedy, they point to splits that respect the dependencies between training and test data (block cross-validation and the like) [Ref. 10, 13]. scikit-learn's GroupKFold is a way of splitting so that the same group never appears in both training and evaluation sets Sourced [Ref. 20].

The following experiment on synthetic data shows how the way of splitting changes the evaluation (Our calculation). We made 30 families of 10 samples each. The features of each family are a fixed pattern for that family plus small noise, and the property value is a family-specific value (standard deviation 2 between families) plus small noise (0.3). That property value is unrelated to the differences in features within a family. Running 5-fold cross-validation with a simple 3-nearest-neighbour predictor gives:

• Random split: coefficient of determination R² = 0.96 (samples of the same family are also in training, so simply copying the values of near neighbours scores high).
• Split by family: R² = −1.28 (the evaluation families are entirely absent from training, and the value for a new family cannot be predicted).

Our calculation. An extreme synthetic dataset (the property is set by the family alone). How far apart the two figures are for real data depends on the size of the family effect. R² can fall below 0 (worse than predicting the mean).

(2) Extrapolation: searching for new materials is usually an extrapolation problem

Meredig et al. state that "traditional machine learning metrics overestimate model performance for materials discovery". They introduced leave-one-cluster-out cross-validation (LOCO CV), which removes a whole cluster of similar materials at a time for evaluation, and a simple nearest-neighbour benchmark. They report that performance in discovery applications depends strongly on the problem, the data sampling and the extrapolation Sourced [Ref. 11]. If the aim is to search outside the range of the training data (a new chemical system, a new composition region), a number from a random split is not an evaluation that fits that aim (this article's commentary).

(3) Small datasets

Zhang and Ling noted that materials datasets are typically smaller, and sometimes more diverse, than in other fields, and studied the relation between data size and predictive performance. They found that the effect of data size is mediated by the degrees of freedom (DoF) of the model, and that the appearance of a precision-DoF association signals underfitting Sourced [Ref. 14]. Matbench, the benchmark for materials property prediction by Dunn et al., has 13 tasks ranging from 312 to 132,000 samples, and its authors report that crystal-graph methods appear to outperform traditional descriptor-based methods given roughly 10⁴ or more data points Sourced [Ref. 12]. The reading (this article's commentary) is that the less data you have, the more the quality of the data, and the effort of building physical and chemical knowledge into the features, count for compared with the choice of model.

(4) Shortcut learning, and different models with the same score

Geirhos et al. tested CNNs and humans on images with conflicting texture and shape cues. They reported a texture bias in ImageNet-trained CNNs and showed that changing the training images could produce shape-based representations [Ref. 15]. This is a result for specific images and models, not a universal law. Our materials-image example is a caution to check whether a model relies on background rather than the specimen. For behaviour that changes across deployment settings, see the original study by D’Amour et al. [Ref. 16].

8. Where compute comes in — parameter count and memory

Small networks, and machine learning on materials features, can often be trained on a laptop in seconds to minutes, so the constraint from computing resources is small (this article's commentary). For models with billions of parameters or more, on the other hand, memory is the first constraint.

The ZeRO paper explains that when training with Adam in mixed precision (computing in half precision, fp16, and keeping an fp32 copy), each parameter needs 2 bytes for the parameter (fp16), 2 bytes for the gradient (fp16) and 12 bytes of optimiser state (the fp32 copy of the parameter, the first moment and the second moment), 16 bytes in all Sourced [Ref. 18]. For a model with Ψ parameters, that is 16Ψ bytes. From this we can compute a lower bound on memory for each model size.

Our calculation. 16Ψ bytes is the mixed-precision Adam case in the ZeRO paper [Ref. 18]. Activations (intermediate values kept for backpropagation), batch size and communication buffers are not included (real requirements are larger). GPU memory capacity differs by product and generation, so no particular product's figure is used here.

This constraint leads to demands on HBM (high bandwidth memory) stacking, on 2.5D packaging such as CoWoS, and on thermal interface materials to carry heat away. We treat these in our explainers on HBM, CoWoS, thermal interface materials (TIM) and AI accelerators.

9. Limits, and what could not be confirmed / Summary

(1) The texts of the standards

The texts of ISO/IEC 22989, 23053, TS 4213, 5259-1, 5259-3, 23894 and 42001 are paid, and we did not check them. This article limits itself to the numbers, titles, publication dates and stages shown on the ISO pages, and states no requirements or definitions.

(2) The numbers from synthetic data

The numbers in Sections 4 to 7 come from calculations on synthetic data that we generated. The errors by degree, the gap between R² values and the confusion-matrix example are there to show mechanisms; they do not indicate the size of the effects in real materials data.

(3) The conditions of universal approximation

For the result of Hornik et al. we kept to the title and the general understanding. We did not check finer details, such as the conditions on the activation function and the sense of approximation, against the body of the original paper.

(4) Methods not covered

We do not cover decision trees, random forests and gradient boosting, support vector machines, Gaussian-process regression, or convolutional, recurrent and graph neural networks. These are probably also widely used on tabular materials data, but we did not check primary sources for that. Transformers are covered in our explainer on Transformers and large language models.

(5) Practical use of reinforcement learning in materials and process work

We did not confirm practical examples of reinforcement learning in materials or process work.

10. How to read evaluations and standards

The performance of machine learning becomes a number only once you fix the evaluation data, how the data are split, the metric and the baseline for comparison. Classification is measured by metrics computed from the confusion matrix, regression by error metrics, and methods are compared on shared research benchmarks (ImageNet, Matbench and others). Safety and reliability need more than a number: standards and guidance (ISO/IEC, NIST) cover data quality, risk management and how an organisation operates. The table below sets out what we could confirm (checked 15 October 2026).

The arrangement of the table is Our calculation (organisation). The numbers, titles, dates and stages of ISO/IEC standards were checked on the official ISO pages (checked 15 October 2026). The texts of the standards are paid, so we limit ourselves to their scope.

Cautions in reading (for materials engineers)

• Numbers change with settings and with who ran the evaluation: R² and accuracy change with the unit of the data split, the order of pre-processing and who evaluated (the developer or a third party). Do not rank figures obtained under different settings against each other just because the metric has the same name.
• An evaluation set wears out with re-use: when the same evaluation set is used for a long time, the numbers can reflect adaptation to that set, or quirks peculiar to how it was made [Ref. 8, 15].
• Most standards and guidance are voluntary: ISO/IEC standards generally gain binding force only when cited in law or contract, and NIST AI RMF is intended for voluntary use [Ref. 28] (the general position of ISO/IEC standards is this article's commentary). When someone says "compliant with a standard", check which requirements of which standard they mean.
• Stated values and measured values are different: an accuracy in a paper or catalogue is a value from a conditioned experiment. It is worth re-measuring on your own data with your own split.

11. Glossary

12. References

1. Rumelhart D. E., Hinton G. E., Williams R. J. "Learning representations by back-propagating errors", Nature 323, 533–536 (1986), DOI 10.1038/323533a0 — nature.com/articles/323533a0
2. Kingma D. P., Ba J. "Adam: A Method for Stochastic Optimization", arXiv:1412.6980 (ICLR 2015) — arxiv.org/abs/1412.6980
3. Hornik K., Stinchcombe M., White H. "Multilayer feedforward networks are universal approximators", Neural Networks 2, 359–366 (1989), DOI 10.1016/0893-6080(89)90020-8 — doi.org/10.1016/0893-6080(89)90020-8
4. Srivastava N. et al. "Dropout: A Simple Way to Prevent Neural Networks from Overfitting", J. Mach. Learn. Res. 15, 1929–1958 (2014) — jmlr.org/papers/v15/srivastava14a.html
5. Zhang C. et al. "Understanding deep learning requires rethinking generalization", arXiv:1611.03530 (ICLR 2017) — arxiv.org/abs/1611.03530
6. Belkin M. et al. "Reconciling modern machine-learning practice and the classical bias–variance trade-off", Proc. Natl. Acad. Sci. USA 116, 15849–15854 (2019), DOI 10.1073/pnas.1903070116 — doi.org/10.1073/pnas.1903070116
7. Nakkiran P. et al. "Deep Double Descent: Where Bigger Models and More Data Hurt", arXiv:1912.02292 — arxiv.org/abs/1912.02292
8. Recht B. et al. "Do ImageNet Classifiers Generalize to ImageNet?", arXiv:1902.10811 (ICML 2019) — arxiv.org/abs/1902.10811
9. Russakovsky O. et al. "ImageNet Large Scale Visual Recognition Challenge", Int. J. Comput. Vis. 115, 211–252 (2015), DOI 10.1007/s11263-015-0816-y — doi.org/10.1007/s11263-015-0816-y
10. Kapoor S., Narayanan A. "Leakage and the reproducibility crisis in machine-learning-based science", Patterns 4, 100804 (2023), DOI 10.1016/j.patter.2023.100804 (open access) — doi.org/10.1016/j.patter.2023.100804
11. Meredig B. et al. "Can machine learning identify the next high-temperature superconductor? Examining extrapolation performance for materials discovery", Mol. Syst. Des. Eng. 3, 819–825 (2018), DOI 10.1039/C8ME00012C — doi.org/10.1039/C8ME00012C
12. Dunn A. et al. "Benchmarking materials property prediction methods: the Matbench test set and Automatminer reference algorithm", npj Comput. Mater. 6, 138 (2020), DOI 10.1038/s41524-020-00406-3 (open access) — doi.org/10.1038/s41524-020-00406-3
13. Roberts D. R. et al. "Cross-validation strategies for data with temporal, spatial, hierarchical, or phylogenetic structure", Ecography 40, 913–929 (2017), DOI 10.1111/ecog.02881 — doi.org/10.1111/ecog.02881
14. Zhang Y., Ling C. "A strategy to apply machine learning to small datasets in materials science", npj Comput. Mater. 4, 25 (2018), DOI 10.1038/s41524-018-0081-z (open access) — doi.org/10.1038/s41524-018-0081-z
15. Geirhos R. et al. "ImageNet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness", ICLR 2019, arXiv:1811.12231 — https://arxiv.org/abs/1811.12231
16. D'Amour A. et al. "Underspecification Presents Challenges for Credibility in Modern Machine Learning", arXiv:2011.03395 — arxiv.org/abs/2011.03395
17. Lakshminarayanan B., Pritzel A., Blundell C. "Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles", arXiv:1612.01474 (NeurIPS 2017) — arxiv.org/abs/1612.01474
18. Rajbhandari S. et al. "ZeRO: Memory Optimizations Toward Training Trillion Parameter Models", arXiv:1910.02054 (SC20) — arxiv.org/abs/1910.02054
19. scikit-learn User Guide, "Cross-validation: evaluating estimator performance" — scikit-learn.org/stable/modules/cross_validation.html
20. scikit-learn API, "GroupKFold" — scikit-learn.org/.../sklearn.model_selection.GroupKFold.html
21. ISO/IEC "ISO/IEC 22989:2022 Information technology — Artificial intelligence — Artificial intelligence concepts and terminology" (ISO page) — iso.org/standard/74296.html
22. ISO/IEC "ISO/IEC 23053:2022 Framework for Artificial Intelligence (AI) Systems Using Machine Learning (ML)" (ISO page) — iso.org/standard/74438.html
23. ISO/IEC "ISO/IEC TS 4213:2022 Information technology — Artificial intelligence — Assessment of machine learning classification performance" (ISO page) — iso.org/standard/79799.html
24. ISO/IEC "ISO/IEC 5259-1:2024 Artificial intelligence — Data quality for analytics and machine learning (ML) — Part 1: Overview, terminology, and examples" (ISO page) — iso.org/standard/81088.html
25. ISO/IEC "ISO/IEC 5259-3:2024 Artificial intelligence — Data quality for analytics and machine learning (ML) — Part 3: Data quality management requirements and guidelines" (ISO page) — iso.org/standard/81092.html
26. ISO/IEC "ISO/IEC 23894:2023 AI — Guidance on risk management" (ISO page) — iso.org/standard/77304.html
27. ISO/IEC "ISO/IEC 42001:2023 AI management systems" (ISO page) — iso.org/standard/81230.html
28. NIST "Artificial Intelligence Risk Management Framework (AI RMF 1.0)", NIST AI 100-1, January 2023, DOI 10.6028/NIST.AI.100-1 — doi.org/10.6028/NIST.AI.100-1

13. Claim-to-source audit

Last updated 15 October 2026. Sources are limited to primary material (peer-reviewed papers, arXiv papers, and ISO and NIST official pages). The texts of ISO/IEC standards are paid and could not be checked. The numerical examples are calculations on synthetic data that we generated, not results on real materials data. The texts of ISO/IEC standards, the finer conditions of the universal approximation theorem, practical use of reinforcement learning in materials work, and other methods such as decision trees could not be confirmed and are not stated here. All figures are explanatory concept graphics or drawings that include our calculations. Figs. 1 and 6 are ImageGen-generated conceptual illustrations, and none of them shows a real cross-section, micrograph or physical product.