The alloy in question, Mg-8wt.%Li, sits in a sweet spot of composition. Above roughly 8 percent lithium by weight, the material becomes dual-phase, containing both the hexagonal close-packed alpha phase typical of magnesium and the body-centered cubic beta phase that lithium stabilizes. That b.c.c. beta phase is the secret ingredient: it deforms far more readily than the hexagonal phase, endowing the alloy with ductility that no other magnesium alloy system can match. The Mg–Li family holds the record for elongation among all magnesium alloys, although its absolute strength remains modest, which is precisely why understanding and controlling its deformation mechanisms matters so much for lightweight structural applications.

To push the alloy into its most deformable state, the team turned to high-pressure torsion, a severe plastic deformation technique in which a disk of material is squeezed under a nominal pressure of 4 gigapascals while being twisted through five full turns. The process shears the microstructure into an ultrafine grain structure. Electron microscopy and transmission Kikuchi diffraction revealed that the extruded material, which started with grains of about 2.2 micrometers, was refined to an average spatial grain size of roughly 0.4 micrometers, with individual subgrains resolved in the range of 0.15 to 0.40 micrometers. The two phases became elongated and aligned with the torsion direction, a signature of the intense shear the disk experienced.

Miniature tensile specimens, cut from the disk mid-radius by electric discharge machining with a gauge cross-section of just 0.8 by 0.8 millimeters, were then pulled at room temperature across strain rates spanning from 10^-5 to 10^-3 per second. The results were striking. The peak stress varied dramatically with strain rate, a hallmark of high strain rate sensitivity, and the alloy exhibited large elongations at the slowest rates of 10^-4 and 10^-5 per second. This unusual increase in ductility at low strain rates, previously reported for similar alloys, is exactly the behavior that makes dual-phase Mg–Li alloys candidates for room-temperature superplasticity, a property normally reserved for ceramics and exotic superalloys at soaring temperatures.

Creep tests added another layer of intrigue. Under constant loads between 10 and 100 megapascals at room temperature, and at 373 kelvin under 10 and 25 megapascals, the creep rates steadily decreased as strain accumulated. Such decelerating creep is normally seen in coarse-grained metals at high temperature, where dislocation substructures build up during the initial stage of deformation, but that explanation should not apply to an ultrafine-grained material whose grains are already smaller than any stable subgrain size. The authors point to a recent study on ultrafine-grained aluminum that attributed similar behavior to strengthening by dynamic recovery, and they adopted the strain rate at 2 percent strain as representative of the as-processed state for further analysis.

The most consequential number to emerge from the creep experiments was the activation energy. Using the temperature dependence of the creep rate between room temperature and 373 kelvin, the team estimated activation energies of 64 kilojoules per mole at 25 megapascals and 77 kilojoules per mole at 10 megapascals. Both values fall well below the 135 kilojoules per mole for self-diffusion and the 92 kilojoules per mole for grain boundary diffusion in pure magnesium. The authors attribute this reduction to the large fraction of the b.c.c. beta phase, which has a lower melting temperature and correspondingly lower diffusion barriers. Reported grain boundary diffusion energies for related alloys, 65 kilojoules per mole for Mg-9%Li and 61 kilojoules per mole for Mg-8%Li-1%Zn, bracket the measurements, and the team settled on 75 kilojoules per mole for their modeling. The low activation energy is a clear signal that thermally activated mechanisms are at work even at room temperature.

To interpret these observations, the researchers deployed a two-pronged computational strategy. First, they applied a recently updated model for grain boundary sliding, the mechanism in which grains slide past one another along their boundaries, which is the canonical explanation for superplastic flow. The model treats the total flow stress as the sum of a threshold stress, capturing other strengthening contributions such as solid solutions, and a grain boundary sliding contribution that depends on shear modulus, temperature, grain size, Burgers vector and the grain boundary diffusion coefficient. A rectified linear function is used as a mathematical constraint to prevent the model from predicting negative stresses when thermal activation is strong. Second, they trained machine learning algorithms on 67 carefully curated data points from the literature, restricted to dual-phase Mg–Li alloys containing no alloying elements other than lithium, with lithium contents between 8 and 12 weight percent, to keep composition effects from contaminating the analysis.

The machine learning lineup included two black-box models, support vector regression and random forest, and two white-box symbolic regression models, one unconstrained and one constrained by dimensional consistency. The black-box models, evaluated with Shapley Additive Explanations, agreed that temperature is the dominant control on flow stress, that strain rate plays an intermediate role, and that grain size matters only marginally. On the training data the algorithms performed well, with mean absolute percentage errors ranging from 11.5 percent for the random forest to 35.3 percent for support vector regression. But when the models were asked to predict the new experimental data generated in this study, the black-box approaches collapsed, producing errors exceeding 250 percent, while the grain boundary sliding model and the dimension-constrained symbolic regression held their ground at roughly 35 percent error.

The deeper story emerged when the team plotted predicted flow stress against grain size, strain rate and temperature and compared the trends with experiment. The experiments revealed a fascinating crossover: for grains larger than about half a micrometer, refining the grains strengthens the alloy in classic Hall–Petch fashion, but below that threshold, further refinement actually softens it, the inverse Hall–Petch regime. Only the grain boundary sliding model captured this transition cleanly. The black-box models failed to reproduce the steep strain rate dependence of the flow stress, with support vector regression even predicting a physically implausible negative strain rate sensitivity at high rates. The dimension-constrained symbolic regression, guided by physically meaningful features such as the Zener–Holomon parameter and dimensionless groups derived from the grain boundary sliding equations, performed far better than its unconstrained counterpart, demonstrating that embedding physical knowledge into machine learning pays real dividends.

The take-home message resonates well beyond magnesium. Machine learning models are only as trustworthy as the data behind them, and with a sparse training set of 67 points, the black-box algorithms interpolated plausibly but extrapolated badly, especially outside the temperature and strain rate ranges they had seen. The century-old physics-based model, by contrast, reproduced every experimental trend once its activation energy was adapted to the dual-phase alloy. For engineers dreaming of lightweight magnesium–lithium components that can be superplastically formed at room temperature, the study offers both a validated design tool and a cautionary tale: in materials science, the white box and the physics still matter, and the smartest models are those that know the laws they are trying to learn.

Subject of Research: Deformation mechanisms and grain boundary sliding in ultrafine-grained dual-phase Mg–Li alloys processed by high-pressure torsion

Article Title: Analysis of deformation mechanism of dual-phase Mg–Li alloy, processed by high-pressure torsion, using experiments and machine learning

Article References: Carvalho, A. P., Edalati, K., & Figueiredo, R. B. (2026). Analysis of deformation mechanism of dual-phase Mg–Li alloy, processed by high-pressure torsion, using experiments and machine learning. Journal of Materials Science. https://doi.org/10.1007/s10853-026-13355-x

Image Credits: AI Generated

DOI: 10.1007/s10853-026-13355-x

Keywords: magnesium-lithium alloy, high-pressure torsion, grain boundary sliding, severe plastic deformation, machine learning, superplasticity, creep, activation energy, Hall-Petch, symbolic regression, ultrafine grains, flow stress

The method, called Equation Discovery for Classification, or EDC, was developed by Guus Toussaint and Arno Knobbe of Leiden University. It extends a research tradition that has long flourished in regression: symbolic regression, the automated search for analytical formulas that fit numerical data. Theauthors of the study point out that equation discovery has historically been applied almost exclusively to problems where the target is a continuous number, such as recovering physical laws like the relationship between the pressure, volume, and temperature of a gas. Their contribution is to redirect that machinery toward binary classification, where the goal is to separate two classes cleanly and explainably.

The core idea is elegantly simple in concept. Rather than learning an opaque scoring function, EDC searches for a concise mathematical expression f(x) and a threshold theta, such that a data point is assigned to the positive class whenever f(x) meets or exceeds theta. For a linear equation, this is essentially the geometry underlying logistic regression or a linear support vector machine: a hyperplane slicing the feature space into two halves. What sets EDC apart is that the search is not confined to straight lines. By allowing nonlinear building blocks such as products of features and exponential terms, the algorithm can trace curved, interaction-driven boundaries while still producing an expression short enough for a domain expert to read, critique, and even correct by hand.

Technically, the framework rests on two pillars: a structured search and a dedicated numerical optimizer. The search space of candidate equations is defined by a configurable context-free grammar, a design choice inherited from classic work on declarative bias in equation discovery. The grammar constrains equations to sums of simple summands, including linear terms, products of two features, and exponential expressions, each parameterized by constants. Crucially, the grammar is redundancy-aware: constructions that would produce syntactically different but semantically identical equations are pruned in advance, since, for example, subtraction between summands is unnecessary when a constant coefficient can simply be negative. Traversing this space exhaustively is impossible for all but the smallest problems, so the algorithm employs beam search, iteratively refining the most promising candidate equations level by level while keeping only a fixed number of survivors at each stage.

Fitting the constants inside each candidate equation proved surprisingly subtle. Although every equation in the grammar is differentiable, gradient descent performed poorly in practice, largely because of the exponential terms that pepper the search space. The authors instead adopted a tailored hill-climbing procedure that first samples a large pool of random constant configurations, then concentrates its remaining budget on refining the best ones. In a systematic comparison against off-the-shelf optimizers from the SciPy library, including Powell, Cobyqa, Cobyla, Nelder-Mead, and stochastic gradient descent, this hill climber achieved the highest mean area under the ROC curve across one hundred randomly generated test problems. Perhaps most strikingly, a simple random-sampling baseline already reached a mean AUC of 0.9974 with a budget of one thousand evaluations, revealing that the inner optimization problem is more tractable than one might fear.

The experiments on synthetic data produced one of the study’s most intriguing findings. When Gaussian noise was injected into datasets whose generating decision boundaries were known, EDC did not merely match the original boundary; beyond a certain noise level it outperformed it. The explanation is a phenomenon the authors describe carefully: noise pushes data points near a curved boundary across it, and concave sections of that boundary collect more stray points than they lose, causing the effective boundary embedded in the data to drift away from its original position and gradually straighten. EDC, fitting the data rather than the hidden formula, tracks this shifted, smoothed boundary, achieving higher AUC than the very equation that generated the data. As noise grew from negligible to substantial across seventeen hundred artificial datasets, even a restricted linear version of EDC eventually beat the original nonlinear boundary.

The framework also proved capable of reconstructing notoriously difficult structures, including XOR-like and interaction-driven boundaries, and of fitting data produced by Gaussian clusters where no explicit target equation exists at all. On these cluster-based problems, which the authors describe as closest to real-world conditions among their artificial experiments, EDC outperformed existing symbolic classification approaches while landing near state-of-the-art black-box methods such as random forests, multi-layer perceptrons, and radial-basis-function support vector machines, all of which posted mean AUC values around 0.97.

Real-world benchmarks reinforced the pattern. Across nine binary classification datasets from the UCI repository, spanning tasks from banknote authentication to income prediction, EDC achieved a higher AUC than every competing equation-discovery-based classifier on every dataset, and beat simple decision trees across the board. A critical distance analysis showed that random forests, neural networks, and SVMs did not statistically significantly outperform EDC, even though those black-box methods won on several datasets, particularly ionosphere and sonar, where feature-class relationships appear to lie outside EDC’s current set of building blocks. On the Adult income dataset, the discovered equation offered a vivid demonstration of interpretability in action: one term acted as a penalty on years of education only when the individual appeared as a child in the household, while an exponential term over marital status effectively functioned as an if-else statement, adding roughly 28,586 to the score for married individuals living with a spouse and a negligible 8 otherwise.

The method’s main weakness is computational cost. In its default configuration, with a search depth of six and a beam width of ten, EDC takes dramatically longer than the sub-second runtimes of conventional classifiers, largely because pairwise interaction terms grow quadratically with the number of features and because one-hot encoding of categorical variables inflates the feature count. The authors show, however, that the expense is largely optional. Restricting the search depth, narrowing the beam, or replacing interaction terms with simple quadratic terms produced speed-ups of up to a factor of fifty at only a marginal loss of accuracy, with no statistically significant difference in performance across the benchmark suite. The authors also note that the framework is not limited to two classes: standard one-versus-rest schemes can extend EDC to multi-class problems, with each of the resulting equations explaining when a particular class label prevails.

The work arrives amid a broader movement toward explainable machine learning, driven by domains such as medicine, finance, and criminal justice where decisions carry real consequences and regulators increasingly demand justification. EDC offers a principled bridge between symbolic regression and classification, delivering models that are simultaneously compact, expressive, and auditable. While it may not dethrone random forests or deep networks on raw accuracy, it demonstrates that the gap is small enough, and the interpretability dividend large enough, that equations once again deserve a seat at the machine learning table. The authors have released their code, experimental setup, and datasets openly so that the entire study can be reproduced and the grammar extended by practitioners in new domains.

Subject of Research: Equation discovery for binary classification using interpretable symbolic decision boundaries

Article Title: Equation Discovery for Classification: Finding Interpretable Symbolic Specifications of the Decision Boundary

Article References: Toussaint, G., & Knobbe, A. (2026). Equation Discovery for Classification: Finding Interpretable Symbolic Specifications of the Decision Boundary. Machine Learning, 115(9), Article 215. https://doi.org/10.1007/s10994-026-07156-1

Image Credits: AI Generated

DOI: 10.1007/s10994-026-07156-1

Keywords: equation discovery, binary classification, symbolic regression, decision boundary, interpretable machine learning, beam search, symbolic classification, explainable AI, context-free grammar, UCI datasets, machine learning, Equation

Constitutive models are the connective tissue of solid mechanics. They are the mathematical rules that translate stress, strain, temperature, and strain rate into predictions of how a real object will bend, dent, stretch, or shatter. Every crash simulation of a car body, every structural analysis of a bridge, and every design study for a battery electrode depends on them. Hao Xu, a postdoctoral researcher at EIT and lead author of the study, explains that the traditional paradigm, while enormously successful, carries an inherent ceiling. Researchers derive a mathematical form based on physical intuition and then calibrate its parameters with data, but the predetermined equation structure restricts what the model can ultimately describe and predict. If the true material behavior involves a functional dependence the chosen formula cannot express, no amount of parameter fitting will recover it. The new framework, called GraphED, inverts the workflow: data comes first, and the equation emerges from it.

The central technical obstacle to such data-driven discovery is representation. Machine learning systems can only optimize what they can encode, and mathematical equations are notoriously awkward objects for algorithms to manipulate. Yuntian Chen, EIT associate professor and co-corresponding author of the study, describes the challenge as efficiently encoding both equation architectures and material-specific parameters into a searchable format for computational algorithms. Conventional symbolic regression approaches typically represent expressions as trees, with operators branching from a root node down to leaves. Trees, however, make it difficult to share structure across different materials and to embed tunable parameters cleanly. The EIT team’s solution is to abandon the tree entirely and encode each candidate equation as a directed graph. In this graph architecture, nodes correspond to mathematical operators and physical variables, while directed edges define the logical and computational connections between them.

The graph representation is more than a cosmetic change, because the edges themselves become carriers of physical meaning. An edge can hold a fixed physical constant, such as a universal exponent, or a tunable material-dependent parameter that varies from one alloy or polymer to another. This design allows GraphED to do two things simultaneously that earlier methods treated separately. It can identify a universal mathematical structure shared across diverse materials and experimental conditions, and it can adaptively calibrate personalized parameters for each individual material scenario. The framework then iterates: it generates candidate graph-structured equations, evaluates how well each one reproduces the experimental data, and optimizes both the topology of the graph and the values of its parameters. The output is a constitutive law that is physically consistent, mathematically compact, and, crucially, interpretable by a human engineer who can read the equation and understand what each term contributes.

That last quality, interpretability, is what separates this work from the black-box models that dominate much of modern machine learning. A neural network trained on material data may predict well, but it offers no equation, no insight, and no transferable knowledge. GraphED delivers the opposite: an explicit formula that a researcher can inspect, question, embed in existing simulation software, and carry into new contexts. The team validated this promise across three very different solid material systems, each chosen for its distinct mechanical character and its historical resistance to empirical modeling. The breadth of the test cases matters, because a discovery method that only works on one material family is a curiosity; one that generalizes across steels, metals, and elastomers is a tool.

The first test case was alloy steel, the workhorse of structural engineering, whose behavior under high strain rates and large plastic deformation is governed by strain-rate dependence and strain hardening. These are precisely the phenomena captured by the Johnson–Cook model, one of the most widely adopted constitutive formulations in impact and crashworthiness analysis. GraphED discovered explicit equations governing both strain-rate dependence and strain hardening directly from experimental data, and when the team integrated these data-driven equations into a complete constitutive model, the result delivered more precise mechanical predictions than the Johnson–Cook benchmark. For a field in which the incumbent model has been refined over decades, the demonstration that an autonomously discovered equation can beat it on predictive accuracy is a significant result.

The second material pushed the method into far harder territory. Lithium metal is a critical component of next-generation energy devices, including high-energy-density batteries, but its mechanical behavior is notoriously difficult to model. Its plastic flow is highly sensitive to both temperature and strain rate, and the interplay of these dependencies has made traditional empirical formulations unreliable. GraphED yielded concise plastic-flow constitutive equations that achieve superior agreement with experimental measurements compared with conventional empirical models. For battery designers, whose simulations of electrode deformation and dendrite-related failure depend on accurate mechanical descriptions, the ability to extract a trustworthy equation directly from data could shorten development cycles and improve the fidelity of electrochemical-mechanical coupled models.

The third validation targeted filled rubbers, a class of elastomers whose hyperelastic response changes with filler composition and temperature. Rubber components in tires, seals, and vibration isolators experience large, reversible deformations, and their stiffness depends on both what they are made of and how hot they are. GraphED captured a compact hyperelastic constitutive equation that maintains robust accuracy across varying material compositions and temperature conditions, meaning a single discovered formula could serve a family of rubber compounds rather than requiring a bespoke model for each formulation. Together, the three case studies span rate-dependent plasticity, temperature-sensitive metal plasticity, and finite-strain hyperelasticity, covering a remarkably wide slice of solid mechanics with one framework.

The researchers see the implications reaching well beyond computational mechanics. Dongxiao Zhang, Chair Professor at EIT and corresponding author of the study, notes that many complex material mechanical behaviors cannot be well described by existing empirical constitutive models, and that equation discovery via graph-based AI provides a powerful new paradigm to assist researchers in deriving rigorous mathematical descriptions when traditional model forms fall short. The phrase that matters is rigorous mathematical description: the goal is not a surrogate that mimics data, but a law, expressed in symbols, that captures the underlying behavior. Because the framework requires only experimental or observational data as input, its potential applications extend to any discipline seeking interpretable physical laws, from soft matter physics and biomechanics to geophysics and beyond, wherever data exists but the governing equation does not.

The study also signals a broader shift in how science may handle the equation-discovery step itself. Symbolic regression and equation learning have been growing fields, but the graph-based encoding developed here resolves a key bottleneck by enabling simultaneous optimization of equation topology and material parameterization, a combination that makes the search tractable for realistic, multi-material datasets. If the paradigm spreads, the role of the mechanician may evolve from proposing formulas to curating data, specifying physical constraints, and interpreting the equations that the algorithm returns. The authors declare no competing interest, and the research was published as a peer-reviewed contribution in Science Advances. For a field built on equations handed down through generations of researchers, the prospect of machines that read raw experimental data and hand back compact, readable laws of material behavior is a genuinely new way of doing physics, and the steels, lithium, and rubber that proved it are only the beginning.

Subject of Research: Graph-based AI equation discovery of constitutive laws in solid materials from experimental data

Article Title: AI discovers interpretable constitutive laws in solids directly from data

Article References: AI discovers interpretable constitutive laws in solids directly from data. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: constitutive models, solid mechanics, graph-based equation discovery, symbolic regression, alloy steels, lithium metal, filled rubbers, machine learning, Science Advances, interpretable AI, strain hardening, hyperelasticity