Michael A. Kraus

Multiaxial strength tests give only a few dozen failure stress states, and choosing which limit surface to fit has long relied on manual comparison and visual checks. Researchers recast that choice as sparse model discovery with differentiable penalties, define the surface by the first root along rays so the equivalent stress is explicitly solvable, and add PBP atoms with learnable shape parameters plus competition-gated mixtures of experts. On concrete data the PBP model's cross-validated error is about a quarter of the monomial library's, and measured strength ratios are reproduced within a few per cent.

Multiaxial strength tests give only a few dozen failure stress states, and choosing which limit surface to fit has long relied on manual comparison and visual checks. Researchers recast that choice as sparse model discovery with differentiable penalties, define the surface by the first root along rays so the equivalent stress is explicitly solvable, and add PBP atoms with learnable shape parameters plus competition-gated mixtures of experts. On concrete data the PBP model's cross-validated error is about a quarter of the monomial library's, and measured strength ratios are reproduced within a few per cent.

Constitutive model discovery now asks which environmental variable drives which mechanism, not merely which candidate terms fit the response. A constitutive model with internal variables is written as a time-unrolled structural causal model, candidate graphs are scored by a weak-form equilibrium gap through the boundary value problem, and climate-chamber protocols count as interventions. On a synthetic perforated plate at about 0.01 px displacement noise, interventional data recover the true graph exactly in 7 of 10 runs, with no attribution error; correlated observational data never do.

Constitutive model discovery now asks which environmental variable drives which mechanism, not merely which candidate terms fit the response. A constitutive model with internal variables is written as a time-unrolled structural causal model, candidate graphs are scored by a weak-form equilibrium gap through the boundary value problem, and climate-chamber protocols count as interventions. On a synthetic perforated plate at about 0.01 px displacement noise, interventional data recover the true graph exactly in 7 of 10 runs, with no attribution error; correlated observational data never do.