Bifidelity material models get a posterior, not just a fit: a sampling hypernetwork runs Langevin trajectories over network weights
Synopsis
Bifidelity modeling uses cheap low-fidelity simulations to supplement scarce high-fidelity experiments, but the usual product is a single deterministic fit that says little about how trustworthy a prediction is. This work extends a single-fidelity Langevin sampling hypernetwork to bifidelity data, using a shared-latent data model that predicts both the low-fidelity response and the high-fidelity correction while a learned drift network generates parameter samples from the posterior score. On a synthetic hyperelastic case, conditional staged training cut the SPD-drift RMSE from 0.075 to 0.024 and the convex-potential drift from 0.081 to 0.017; on experimental and DNS data, predicting the high-fidelity response directly gave lower error than an additive discrepancy.
Interpretation
A constitutive material model can now come with a parameter posterior rather than only a calibrated deterministic network. The earlier single-fidelity Langevin sampling hypernetwork handled one fidelity of data and could not use abundant cheap low-fidelity simulations to constrain scarce high-fidelity experiments. On a synthetic hyperelastic case, conditional staged training cut the SPD-drift RMSE from 0.075 to 0.024 and the convex-potential drift from 0.081 to 0.017, with the predictive distribution covering the data and block alternation converging in one sweep.
The mechanism is a sampler network that generates weight samples for a data model shaped like a Y, with a shared input encoding that splits into low-fidelity and high-fidelity branches. The sampler uses Langevin dynamics, and its drift network learns the score of the posterior, the gradient of the log posterior with respect to parameters, pushing parameter trajectories toward regions the data support. The demonstrations cover a synthetic Yeoh hyperelastic case open to cross-validation and paired experimental and bonded-particle-method DNS data for a direct-ink-write printed lattice material.
The choice of drift architecture directly affects convergence and uncertainty quality, and the convex-potential drift gave consistently lower ELBO and prediction error on the synthetic case. A commonly used U-net-style drift carries no dissipation guarantee, whereas a symmetric positive-definite linear drift and an input-convex neural network potential drift guarantee decay to a steady state. In convergence comparisons using half the available high-fidelity data, the convex-potential drift kept the lowest ELBO and prediction metrics and its weight distributions reached steady state earlier.
Perspective
The demonstrated scope is one synthetic Yeoh hyperelastic case and one direct-ink-write printed lattice material case combining experiments with DNS, the latter keeping a single design variable, filament spacing, and splitting training and test data by contiguous trajectory blocks rather than by experiment. The intended readers are researchers and engineers who need uncertainty-aware surrogates for scarce high-fidelity data.
In the real material case the high-fidelity response changes stiffness abruptly near a given strain with no counterpart in the low-fidelity DNS, and the response is monotonic but not convex, so no polyconvexity constraint was imposed. The precision spectrum from Langevin sampling is flatter than that from Hamiltonian Monte Carlo, underestimating dominant modes and overestimating weakly constrained ones, which makes it a conservative estimate; Hamiltonian Monte Carlo is more faithful but far more expensive. The authors also note the weight posterior is likely multimodal while the drift models assume unimodality.