Simulation-Based Inference of Ginzburg--Landau Parameters in Type--1.5 Superconductors

Abstract

Inferring microscopic couplings in multi-component superconductors directly from vortex configurations is a challenging inverse problem. In Type-1.5 systems, Time-Dependent Ginzburg-Landau (TDGL) dynamics generate complex, glassy vortex patterns with high metastability. We explicitly quantify this intractability by analyzing the Hessian spectrum of the energy landscape, revealing a proliferation of soft modes that hinders traditional sampling. We address this challenge by combining a differentiable TDGL solver with Simulation-Based Inference (SBI). Our approach treats the solver as a stochastic forward model mapping physical parameters (θ = (η, B, ν)) to vortex density fields. Using Neural Ratio Estimation (NRE), we train a classifier to approximate the likelihood-to-evidence ratio and perform Bayesian inference for the interband Josephson coupling from vortex density fields. On synthetic data, the proposed method reliably recovers the coupling with calibrated uncertainty.

Figures (7)

• Figure 1: Overview of the Framework. The pipeline integrates differentiable physics simulation with neural ratio estimation. (Phase 1) A TDGL solver generates vortex configurations from parameters $\boldsymbol{\theta}=(\eta, B, \nu)$. The forward model is augmented with two physical diagnostic loops (dashed lines): Hessian spectral analysis to quantify the ruggedness of the energy landscape (see Sec. \ref{['sec:hessian']}), and structure-factor $S(k)$ analysis to reveal the micro-structural fingerprint of the drag mechanism (see Sec. \ref{['subsec:drag_discovery']}). (Phase 2) A contrastive dataset is constructed on-the-fly by pairing consistent $(\boldsymbol{\theta}, x)$ samples (joint) against permuted samples (marginal). (Phase 3 & 4) A two-tower neural architecture fuses visual vortex features with parameter embeddings to approximate the likelihood-to-evidence ratio, trained via binary cross-entropy maximization.
• Figure 2: Posterior Recovery across Coupling Strengths. The predicted posterior distributions for five distinct ground-truth values of $\eta$. The model accurately localizes the true parameter (red dashed line) within the high-probability region. Note the widening of the posterior at $\eta=1.4$, correctly reflecting the increased physical ambiguity in the strong-coupling limit.
• Figure 3: Joint Posterior Distribution $p(\eta, B | x)$. Heatmap of the predicted posterior probability for a test case with $\eta=0.8$ and $B=0.04$. The posterior shows a strong localization along the $\eta$-axis (coupling strength), confirming that the NRE successfully identifies the microscopic interaction mechanism from the vortex clustering patterns. In contrast, the posterior is broader along the $B$-axis (magnetic field). This increased uncertainty reflects the non--equilibrium nature of the rapid quench simulations: the final vortex density is significantly influenced by the freezing dynamics (Kibble--Zurek mechanism) and does not always strictly correspond to the equilibrium external field $B$. The model correctly captures this intrinsic physical ambiguity while still localizing the ground truth (red star) within the high-probability region.
• Figure 4: Dynamical Fingerprint of the Drag Effect. Temporal evolution of the structure factor intensity at low wavevector ($S(k \approx 1)$) plotted on a logarithmic scale. During the early relaxation phase ($t=1.0$, highlighted), the model with Andreev-Bashkin drag (red) exhibits a distinct suppression of density fluctuations compared to the standard model (blue), indicating a transient tendency towards hyperuniformity driven by kinetic locking. As the system evolves towards metastability ($t \ge 3.0$), the curves converge, confirming that the drag signature is dynamically encoded in the formation pathway rather than the final static equilibrium.
• Figure 5: Diagnostic Analysis. (Left) Parameter recovery error (MAE) as a function of $\eta$, showing a "sweet spot" near $\eta=0.8$. (Right) The width of the credible intervals. The 95% CI (purple) widens significantly at strong coupling ($\eta=1.4$), indicating that the model correctly identifies the reduced information content in that regime.