materials-chemistry/ai produced the result/Materials 2023 · v2

Neural network built from physics equations extracts solder deformation coefficients

Researchers rewrote three equations describing how lead-free solder deforms as neural networks whose internal weights are the equations' own coefficients. Training on published measurements, then refining with Bayesian statistics, produced values for those coefficients.

spectrum · one line per step, placed by what the step does · bright lines used AI

Coefficient Extraction of SAC305 Solder Constitutive Equations Using Equation-Informed Neural Networks
Materials, 2023

doi:10.3390/ma16144922 · record aix-00033 v2 · checked 2026-10-08

ai-resultrole of AI

AI was for: Property prediction
Model family: Multilayer perceptron, Probabilistic graphical model
Checked by: Replication
Code: not reported

The finding the paper is about came from the AI.

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The science is explained before the AI appears. Switch to field specialist to go straight to the method.

Assumes the discipline and goes straight to the method.

The diagram, the record and what the paper did not report are identical in both modes. Only the framing changes — never the evidence.

Introduction by AIxSci · plain language

What this research was about

Solder is the metal that holds electronic components onto circuit boards. Under heat and mechanical load it slowly stretches and creeps, and engineers need to predict that behaviour before a joint cracks. They do so with constitutive equations: formulae that relate the stress in a material to how fast it is being strained and how hot it is. Each formula contains a handful of constants, and those constants differ for every alloy. Finding them means fitting the formula to laboratory measurements, which is awkward because the equations are non-linear and the coefficients can trade off against one another, so several different sets of values may fit the same data.

The researchers took this fitting problem for SAC305, a lead-free tin-silver-copper solder, and recast it as a learning problem. They worked with three established formulae: the hyperbolic Garofalo creep law, the nine-parameter Anand model and the Chaboche model.

Where AI came in

A neural network is a graph of simple arithmetic units joined by connections, each carrying an adjustable number called a weight; training nudges those weights until the network reproduces known input-output pairs. Here the graph was not generic. Each unit was made to compute one term of the constitutive equation, so that the weights were the physical coefficients themselves. Creep and tensile data for SAC305, taken from two earlier published studies, were rescaled and paired up as inputs and outputs. A grid search supplied starting values, then the network was trained by backpropagation, the standard method of adjusting weights from the error at the output. Converting the trained weights back gave the coefficients.

The trained values were then used as the starting point for Bayesian inference, a statistical procedure that returns a spread of plausible values rather than a single answer, sampled here by Markov chain Monte Carlo. The average of each spread became the refined coefficient. The network and the sampling together stand in for the least-squares fitting that is conventionally used to pull coefficients out of such measurements. For the Chaboche case, the refined Anand coefficients were used to generate stress-strain curves at a fixed strain rate, and those curves became the training data for a further network.

Written by AIxSci from the checked record below, to give context for readers outside the field. It is not part of the record.

The work

Technical · from the record

The authors built a neural network whose graph encodes a material constitutive equation and whose weights are that equation's coefficients, so that training the network on measured data pairs yields the coefficients. They applied it to the hyperbolic Garofalo, nine-parameter Anand and Chaboche models for lead-free SAC305 solder, using creep and stress-strain data from two earlier studies, then ran Markov chain Monte Carlo Bayesian inference from the trained values to obtain coefficient distributions. For the Garofalo model, 1000 Bayesian inference iterations were run, and the paper reports that 58 instances of the coefficient C1 differed from the average by more than 5%. Mean square errors of the extracted coefficients are reported alongside those of the coefficients published in the source studies.

How AI was used

Experimental data pairs taken from published SAC305 studies were rescaled by pre-processing mapping functions into a bounded domain, with per-datapair ratios assigned to emphasise low strain rates and temperatures near electronic-component operating conditions. Each constitutive equation (hyperbolic Garofalo, the two steps of the nine-parameter Anand model, and Chaboche) was rewritten as a neural network in which neurons implement the equation's terms and the coefficients serve as weights. Initial coefficient values were located by grid search, with bisection used for one Garofalo coefficient, and selected coefficients were constrained positive. The network was then trained by steepest-descent backpropagation over the weighted data pairs, and post-processing conversion functions mapped the learned weights back to physical coefficients. Those coefficients were used as starting values for a numerical Bayesian inference scheme with gamma-gamma conjugate updates, numerical integration and Gibbs sampling within MCMC, giving a distribution for each coefficient whose mean became the refined value. For the Chaboche case, the Bayesian-refined Anand coefficients were used to generate temperature-dependent stress-strain curves at a fixed strain rate, and those generated curves formed the training set for the Chaboche EINN.

The shape of the work

Structural · the record, drawn

AI stepNo AI↤ what the AI stood in for

1Acquisition
no AI

Collect published experimental datasets for SAC305

Obtaining raw data, whether by measurement, download or retrieval.

The experimental dataset is drawn from Xiao and Armstrong.where the paper describes this · verbatim

in the paper

2Preparation
no AI

Pre-process mapping of data pairs and assign learning ratios

Cleaning, filtering, normalising or labelling data already obtained.

In order to proportional convert the original data to the [a,b+a] domainwhere the paper describes this · verbatim

in the paper

3Representation
no AI

Convert constitutive equations into EINN graph form

Encoding data into features, descriptors, embeddings or graphs.

the foundational concept of EINNs begins with constructing an artificial neural network to embody the constitutive equation fwhere the paper describes this · verbatim

in the paper

4Optimisation
no AI

Grid search for initial coefficient values

Iterative search over a space.

A grid search technique is employed to identify optimal initial values concerning the experimental data.where the paper describes this · verbatim

in the paper

5Training
AI

Train EINN by backpropagation and convert weights to coefficients

Fitting model parameters, including fine-tuning an existing model. The AI stood in for statistical model.

This neural network can be incrementally trained using input and output data pairswhere the paper describes this · verbatim

in the paper

6Optimisation
AI

Refine coefficients with numerical Bayesian inference

Iterative search over a space.

A total of 1000 Bayesian Inference interactions were performed to obtain the distribution of the extracted coefficients.where the paper describes this · verbatim

in the paper

7Simulation
no AI

Generate stress-strain curves from the extracted Anand coefficients

Numerical or physics simulation, including where a learned surrogate replaces it. Its result feeds back into an earlier step.

Stress–strain curves can be generated by the Anand model (as Equations (14) and (15)) for each temperature pointwhere the paper describes this · verbatim

in the paper

8Validation
no AI

Compare coefficient fit against the originating studies

Testing outputs against ground truth.

The MSE values indicate that the coefficients obtained from the EINN formulation exhibit similar accuracy to those obtained using conventional methods.where the paper describes this · verbatim

in the paper

What the record says

Technical · every part carries its own basis

+ in the paper~ our reading− not reported

How to read the quotations. A quotation shows where the paper describes something. It does not quote every value beside it: one passage locates a part of the work, and values without their own quotation are our reading of that passage.

~Role of AI

AI produced the resultour reading

The reported result is the set of constitutive-model coefficients, and those coefficients are produced by training the equation-structured neural network and then refining it with Bayesian inference; without the model there is no result.

~What the AI was for

Property predictionour reading

the corresponding neural network representations of Yi are formulated, and the coefficient Ak is assigned as the weightingwhere the paper describes this · verbatim

~Model families

Multilayer perceptron Probabilistic graphical modelour reading

~How it was taught

Supervisedour reading

~Models named

Equation-Informed Neural Network (EINN) · Trained from scratchNumerical Bayesian Inference model (MCMC with Gibbs sampling) · Trained from scratchour reading

~How results were checked

Replicationour reading

the mean square error (MSE) of the EINN formulation learning was comparable to those from the literaturewhere the paper describes this · verbatim

−Code · weights · data

code not reportedweights not reporteddata not reportednot reported

−Compute

not reportednot reported

What this paper did not report

Technical · absence is published deliberately

Reported as not stated — 8 items

CodeWhether the code is available is not stated.
Trained model weightsWhether the trained model is available is not stated.
DataWhether the data are available is not stated.
ComputeThe hardware or time used is not stated.
How many were testedThe paper gives no count of what was tested.
Version of Equation-Informed Neural Network (EINN)Which version of the model was used is not stated.
Version of Numerical Bayesian Inference model (MCMC with Gibbs sampling)Which version of the model was used is not stated.
What step 6 replacedThe paper gives no basis for what the AI stood in for.

About this article

Record aix-00033, version 2, checked by a person on 2026-10-08. The record describes the paper; it does not assess whether the paper's findings are right. The paper is published under CC-BY; quotations are at most 25 words. How we work · Report an error