Diffusion-Based Impedance Learning

• Diffusion-based impedance learning is an umbrella term for methods that use diffusion processes to invert impedance measurements and extract latent physical and material parameters.
• It integrates drift–diffusion simulations, generative diffusion models, and physics-based regularization to tackle ill-posed inverse problems in applications like solar cells, EIT, and robotics.
• The approach leverages diffusion priors to enhance parameter recovery, improve noise robustness, and adapt to diverse system models through hybrid data-driven and first-principles methods.

Searching arXiv for papers on diffusion-based impedance learning and closely related impedance inversion topics. Diffusion-Based Impedance Learning denotes a family of methods that use diffusion processes—either physical diffusion models, generative diffusion models, or both—to infer latent physical structure, material parameters, conductivity fields, or control-relevant impedance parameters from impedance-related measurements. Across the literature, the term spans several distinct but technically connected settings: drift–diffusion inversion of perovskite solar-cell impedance spectra (Nabil et al., 10 Nov 2025), conditional diffusion models for Electrical Impedance Tomography (EIT) (Shi et al., 2024, Shi et al., 10 Jan 2025), graph-based posterior sampling on finite-element meshes for nonlinear EIT (Alberti et al., 19 May 2026), diffusion-regularized EIT with implicit neural representations (Tong et al., 2024), conditional diffusion for completing undersampled Dirichlet-to-Neumann data (Chen et al., 8 Feb 2026), and diffusion-based learning of variable impedance or stiffness in contact-rich manipulation (Aburub et al., 2024, Geiger et al., 24 Sep 2025, Okada et al., 2024). In another established usage, “diffusion-based” refers to electrochemical or transport-theoretic impedance learning, where impedance features are interpreted through bounded diffusion, drift–diffusion, or diffusion-time distributions to extract transport parameters and microstructural statistics [(Hallemans et al., 2024); (Khazimullin et al., 2018); (Boukraa et al., 2022); (Song et al., 2017); (Kulikovsky, 2021); (Song et al., 2012)]. The common premise is that impedance observables are indirect, often ill-posed encodings of transport, structure, or interaction, and that diffusion-based priors or diffusion-governed forward models provide the inductive bias required for stable inversion.

1. Conceptual scope and problem classes

Diffusion-Based Impedance Learning appears in at least three technically distinct regimes. In the first, diffusion is part of the physical forward model, and learning is used to invert the mapping from impedance to physical parameters. The perovskite-solar-cell study “Inversion of the impedance response towards physical parameter extraction using interpretable machine learning” uses drift–diffusion simulations to generate synthetic impedance spectra and then learns regressors from equivalent-circuit features to bulk lifetimes, ion density, ion mobility, and surface recombination velocities (Nabil et al., 10 Nov 2025). Battery-model parametrisation from EIS follows the same principle, but with porous-electrode diffusion and electrolyte transport in the SPMe rather than ionic–electronic drift–diffusion in perovskites (Hallemans et al., 2024).

In the second regime, diffusion models are generative priors for inverse problems involving impedance data. This is most developed in EIT, where conditional diffusion reconstructs conductivity images from boundary voltages (Shi et al., 2024, Shi et al., 10 Jan 2025), graph diffusion priors are combined with posterior sampling directly on triangular meshes (Alberti et al., 19 May 2026), and diffusion regularizers are coupled to implicit neural representations (Tong et al., 2024). Related work treats diffusion as a preprocessing model for completing missing Dirichlet-to-Neumann measurements before passing them to an inverse solver (Chen et al., 8 Feb 2026).

In the third regime, impedance refers to robot compliance rather than electrical impedance. Here diffusion models learn motion and impedance parameters jointly, or reconstruct latent equilibrium trajectories from contact-perturbed motion and wrench histories, after which stiffness and damping are adapted online (Aburub et al., 2024, Geiger et al., 24 Sep 2025). A related line uses denoising diffusion to model contact trajectories as a surrogate for robot-intensive variable-stiffness optimisation (Okada et al., 2024).

A plausible implication is that “Diffusion-Based Impedance Learning” is best understood as an umbrella term rather than a single algorithmic template. Its unifying structure is an inverse map from impedance-associated observations to hidden quantities, regularized either by physical diffusion laws or by score-based diffusion priors.

2. Physics-based inversion of impedance spectra

In perovskite solar cells, impedance spectra reflect tightly coupled ionic and electronic subsystems, so direct parameter extraction from experiments is difficult. The cited work models a planar n–i–p stack, Glass/FTO/compact TiO2/MAPI/Spiro-OMeTAD/Au, under 1-sun AM1.5G illumination, and solves Poisson, electronic continuity, carrier drift–diffusion, and ionic drift–diffusion equations to generate synthetic EIS data (Nabil et al., 10 Nov 2025). The impedance is defined from a small sinusoidal perturbation as

$$Z(\omega)=\frac{\tilde V(\omega)}{\tilde I(\omega)},$$

with characteristic frequencies related to time constants by

$f_c=\frac{1}{2\pi\tau}.$

The study extracts generalized equivalent-circuit features or direct signal features and trains multi-output regressors for six parameters: bulk electron pseudolifetime, bulk hole pseudolifetime, mobile ion density, ion mobility, and two minority-carrier surface recombination velocities (Nabil et al., 10 Nov 2025).

The reported results show that a Gradient Boosting Regressor using equivalent-circuit fitting features performs best. For open-circuit impedance with equivalent-circuit features, the reported $R^2$ values are 0.80, 0.84, 0.82, 0.68, 0.63, and 0.50 for the six targets; for short-circuit impedance with equivalent-circuit features they are 0.73, 0.69, 0.86, 0.86, $-0.24$, and 0.94 (Nabil et al., 10 Nov 2025). Interpretability via SHAP links high-frequency resistance and capacitance features to recombination lifetimes and surface recombination velocities, and low-frequency features to ionic mobility (Nabil et al., 10 Nov 2025). Open circuit is reported as most informative for recombination losses, whereas short circuit is best for ionic parameters and the spiro-contact recombination velocity (Nabil et al., 10 Nov 2025).

Experimental validation on approximately 16% PCE TiO2/MAPI/spiro devices produced ion concentrations of $(1.3$–$3.3)\times10^{17}\,\mathrm{cm}^{-3}$, ion mobilities of $(5$–$7)\times10^{-11}\,\mathrm{cm^2\,V^{-1}\,s^{-1}}$, and surface recombination velocities of approximately $7$–$9\,\mathrm{m\,s^{-1}}$ at TiO2/MAPI and $f_c=\frac{1}{2\pi\tau}.$0–$f_c=\frac{1}{2\pi\tau}.$1 at MAPI/spiro (Nabil et al., 10 Nov 2025). The paper explicitly notes that equivalent-circuit features carry more learnable information than minimalist peak-based features, and that stack specificity limits direct transfer to other architectures without retraining (Nabil et al., 10 Nov 2025).

Electrochemical impedance inversion in batteries is structurally analogous but grounded in porous-electrode diffusion. The PyBaMM-EIS work computes frequency-domain impedance from any PyBaMM model using automatic differentiation, identifies the SPMe as a parsimonious model that reproduces the electrolyte diffusion “bump,” and fits 18 grouped parameters across multiple states of charge (Hallemans et al., 2024). The linearized frequency response is written as

$f_c=\frac{1}{2\pi\tau}.$2

and diffusion signatures are interpreted through finite-length and semi-infinite transport expressions such as

$f_c=\frac{1}{2\pi\tau}.$3

The study reports fitting errors below 1% across nine SOCs for simulated SPMe data and measured-LG-M50LT fitting errors of approximately 1.78–5.79% across SOC 20–80% (Hallemans et al., 2024). It explicitly concludes that simultaneous fitting across SOC is crucial for identifiability (Hallemans et al., 2024).

Earlier impedance literature already established that bounded diffusion, geometry, and dielectric layers materially affect parameter recovery. The dielectric-layer analysis for blocking-electrode electrolytes shows that neglecting dielectric-layer contributions leads, in most cases, to incorrect diffusion coefficient and ion concentration estimates, and reports roughly threefold diffusion-coefficient overestimation when dielectric layers are ignored in 5CB cells (Khazimullin et al., 2018). Related work on bounded diffusion in microfluidic channel-electrode architectures implements a finite-length Warburg in PyEIS and fits diffusion-layer thickness as a function of flow (Boukraa et al., 2022). The “Distribution of Diffusion Times” framework generalises such interpretations by recovering a distribution over finite-length Warburg or Gerischer elements, enabling “impedance imaging” of heterogeneous electrodes (Song et al., 2017). For insertion electrodes, geometry-aware bounded-diffusion models show that overlooking curvature can overestimate diffusivity by more than $f_c=\frac{1}{2\pi\tau}.$4, while overlooking size distribution can underestimate it by about 16% in silicon nanowires (Song et al., 2012).

3. Conditional diffusion models for electrical impedance tomography

In EIT, the inverse problem is to recover conductivity fields from boundary voltage data governed by the elliptic PDE

$f_c=\frac{1}{2\pi\tau}.$5

with Complete Electrode Model boundary conditions (Shi et al., 2024, Shi et al., 10 Jan 2025, Alberti et al., 19 May 2026, Tong et al., 2024, Wang et al., 2023). Because the inverse map is nonlinear and severely ill-posed, conditional diffusion models have been proposed as learned priors over conductivity fields.

The CDEIT model formulates a conditional denoising diffusion process in which clean conductivity images are progressively noised and then reconstructed conditioned on voltage-derived “electrical impedance maps” (Shi et al., 2024). The forward diffusion follows the DDPM form

$f_c=\frac{1}{2\pi\tau}.$6

with reverse transitions conditioned on boundary voltages $f_c=\frac{1}{2\pi\tau}.$7 through a Transformer-based U-Net (Shi et al., 2024). The reported configuration uses patch size 2, model dimension 512, $f_c=\frac{1}{2\pi\tau}.$8 diffusion steps, $f_c=\frac{1}{2\pi\tau}.$9 linearly increased from $R^2$0 to $R^2$1, Adam with learning rate $R^2$2, batch size 64, and 150k iterations; inference uses DDIM with 5 reverse steps (Shi et al., 2024). The method includes normalization procedures for geometry, current amplitude, and background conductivity so that simulation-trained models can be applied to real datasets (Shi et al., 2024).

On the synthetic test set, CDEIT reports PSNR $R^2$3 dB, SSIM $R^2$4, and CC $R^2$5, outperforming the listed baselines (Shi et al., 2024). On noisy data, performance degrades but remains strong at 40 dB noise, where PSNR is reported as 35.58 dB (Shi et al., 2024). The model has approximately 107.5M parameters and 1.55B FLOPs per forward pass, with about 43.3 hours of training time on an NVIDIA Tesla V100 and fast DDIM inference using 5 steps (Shi et al., 2024).

CDMVC introduces a different conditioning strategy: it uses a physics-based pre-imaging module based on PDIPM with TV regularization, then conditions a diffusion model on that reconstruction, and finally adds a learned forward voltage constraint during sampling (Shi et al., 10 Jan 2025). Its forward model uses the static conductivity equation and time-difference EIT, while the reverse diffusion is based on a VP-SDE/DDPM/DDIM formulation with a U-Net denoiser (Shi et al., 10 Jan 2025). The crucial addition is a forward voltage constraint network $R^2$6 and a sampling-time projection step minimizing

$R^2$7

On simulation data, CDMVC reports RE 0.0634, SSIM 0.9819, PSNR 38.4498, MSE 0.0007, CC 0.9934, and DR 1.0059, improving on CDM without voltage consistency and on a range of classical and GAN baselines (Shi et al., 10 Jan 2025). Runtime is reported as 0.485 s, far faster than the compared prior diffusion-based CSD* at 8.222 s (Shi et al., 10 Jan 2025).

A different conditioning strategy is used in Diff-INR, where a pre-trained diffusion model acts as a generative regularizer on a pixelized conductivity image produced by an implicit neural representation. The conductivity is represented continuously as $R^2$8, and the reconstruction objective combines data fidelity with a score-distillation-style diffusion penalty (Tong et al., 2024). This decouples the representation from any specific inverse mesh and yields robustness across mesh densities (Tong et al., 2024). The paper reports state-of-the-art reconstruction accuracy among self-supervised baselines on both simulation and experimental saline-tank data (Tong et al., 2024).

The comparative study of VAEs, CNFs, and score-based diffusion in EIT shows that no single generative approach dominates under all conditions, but that conditional score-based diffusion demonstrates the best generalization in high-noise settings (Wang et al., 2023). That study’s CSD* uses an unconditional NCSN++ prior with Gauss–Newton initialization inserted into the reverse trajectory, rather than a directly conditioned score model (Wang et al., 2023). This suggests that diffusion-based impedance learning in EIT spans a continuum from fully conditional generative inversion to posterior refinement driven by learned unconditional priors.

4. Posterior sampling, data completion, and mesh-native diffusion priors

A major limitation of image-grid diffusion in EIT is the mismatch between regular image lattices and finite-element meshes. Diffusion Graph Posterior Sampling addresses this by training a graph-native unconditional diffusion prior directly on triangular meshes and then combining it with likelihood guidance through the nonlinear forward operator (Alberti et al., 19 May 2026). The EIT likelihood is written as

$R^2$9

with posterior guidance inserted during reverse sampling via

$-0.24$0

The regularized RDPS variant augments the diffusion prior with explicit TV or generalized Tikhonov penalties on the graph (Alberti et al., 19 May 2026).

The reported experiments use a unit-disk 2D EIT setting with 32 electrodes and meshes of 3766 forward vertices and 1602 inversion vertices (Alberti et al., 19 May 2026). DDIM posterior sampling outperforms DDPM posterior sampling, with mean RMSE 0.0726 versus 0.0764 on a 50-sample test set, and explicit TV regularization further improves mean RMSE from 0.0622 without regularization to 0.0488 (Alberti et al., 19 May 2026). On noisy data, RDPS remains robust, with mean RMSE 0.0708 and 0.0850 under two AWGN levels and 0.0706 and 0.0814 under two Laplacian-noise levels (Alberti et al., 19 May 2026). The method is also reported to generalize to out-of-distribution horseshoe inclusions and to real 2D EIT data without retraining (Alberti et al., 19 May 2026).

Data Completion for Electrical Impedance Tomography moves the generative target from conductivity images to the measurement operator itself (Chen et al., 8 Feb 2026). Rather than reconstructing $-0.24$1 directly from severely undersampled measurements, it learns a conditional diffusion model over Dirichlet-to-Neumann matrices and uses the completed matrix as input to an off-the-shelf inverse solver (Chen et al., 8 Feb 2026). Under the stated polygon-conductivity assumptions, the paper derives a nonasymptotic total-variation bound on the discrepancy between completed and ground-truth DtN data (Chen et al., 8 Feb 2026). Empirically, diffusion completion at 1% sampling achieves relative error 0.9% on the completed off-diagonal block, whereas matrix completion requires roughly 30% random sampling to achieve similar quality (Chen et al., 8 Feb 2026).

End-to-end reconstruction results confirm that completion materially improves downstream inversion. On the “Disks” dataset, the full-data inverse solver achieves SSIM 0.819 and RE 20.7% in the noiseless setting, while diffusion completion at 1% sampling followed by the same solver yields SSIM 0.774 and RE 33.4%, substantially better than a direct sparse-data solver at 1% sampling with SSIM 0.597 and RE 44.9% (Chen et al., 8 Feb 2026). On Shepp–Logan data, diffusion completion at 1% sampling yields SSIM 0.910 and RE 10.3% in the noiseless setting, close to the full-data baseline of SSIM 0.930 and RE 8.2% (Chen et al., 8 Feb 2026).

These works jointly indicate that diffusion-based impedance learning in EIT is not restricted to image-space reconstruction. It also includes posterior sampling on mesh-native state spaces and generative completion of boundary measurement operators.

5. Diffusion-based learning of compliance and impedance in robotics

In robotics, the “impedance” being learned is mechanical rather than electrical, but the inverse-learning structure is analogous: observations of motion, force, and contact are mapped to latent stiffness, damping, or equilibrium trajectories. DIPCOM defines actions as absolute Cartesian end-effector pose, gripper width, and a diagonal Cartesian stiffness vector $-0.24$2, and learns a diffusion policy conditioned on RGB wrist images, wrist force/torque, and Cartesian state history (Aburub et al., 2024). The action loss is

$-0.24$3

and the policy predicts long action horizons of roughly 48 actions to support repetitive contact-rich tasks (Aburub et al., 2024).

Task results show substantial gains over the Comp-ACT baseline. In powder grinding, human demonstrations achieve 76.67% average fine-powder production, DIPCOM achieves 55.88%, and Comp-ACT 9.96% (Aburub et al., 2024). In pencil erasing, DIPCOM achieves 77.32% erased on average with a 52.3% success rate, whereas Comp-ACT achieves 26.0% erased with 0% success (Aburub et al., 2024). For bimanual insertion, both methods achieve 100% success on round pegs and 95% on cuboid pegs, but DIPCOM exhibits more adaptive behaviour (Aburub et al., 2024). The paper explicitly attributes these improvements to better multimodal long-horizon modeling and to force-conditioned compliance control (Aburub et al., 2024).

The DCM work uses denoising diffusion not as a policy but as a surrogate contact model for variable-impedance optimisation (Okada et al., 2024). It predicts contact force trajectories conditioned on demonstrations, attractor trajectories, and candidate stiffness profiles, then uses these predictions inside a multi-objective Bayesian-optimization loop over stiffness (Okada et al., 2024). On simulated wiping tasks, diffusion-based virtual optimisation achieves hypervolume comparable to robot-based optimisation while requiring far fewer real trials; on the real robot, DCM-based robot-free optimisation requires about 25 minutes total compared with about 80 minutes for robot-based optimisation, while achieving comparable performance envelopes (Okada et al., 2024).

DBIL further separates information and energy domains. A Transformer-based diffusion model reconstructs a simulated Zero-Force Trajectory from pose and wrench histories, and an energy-based estimator updates stiffness and damping online (Geiger et al., 24 Sep 2025). The paper introduces a SLERP-based quaternion noise scheduler for rotational diffusion,

$-0.24$4

and uses this reconstructed equilibrium to modulate task-space impedance directionally (Geiger et al., 24 Sep 2025). Reported trajectory reconstruction errors are 0.994 mm and $-0.24$5 on the parkour dataset, improving to 0.883 mm and $-0.24$6 after retraining on combined datasets (Geiger et al., 24 Sep 2025). In deployment, the method achieves smooth parkour traversal within force and velocity limits and 30/30 success on cylindrical, square, and star peg insertions without peg-specific demonstrations (Geiger et al., 24 Sep 2025).

A plausible implication is that these robotic works broaden the term “impedance learning” from passive identification of transport parameters to active learning of interaction laws. What remains invariant is the use of diffusion processes to regularize or generate latent quantities that are otherwise hard to infer from sparse or multimodal observations.

6. Limitations, trade-offs, and recurring methodological themes

Several limitations recur across the literature. Stack specificity is explicit in the perovskite inversion study: the learned mapping depends on architecture, energy alignment, and fixed physical priors, so transfer to other stacks requires retraining or domain adaptation (Nabil et al., 10 Nov 2025). In EIT, real-to-simulation gaps arise from geometry, contact impedances, current amplitude, background conductivity, and measurement noise; different works address these with normalization (Shi et al., 2024), voltage-consistency guidance (Shi et al., 10 Jan 2025), mesh-native priors (Alberti et al., 19 May 2026), or generative regularization coupled to mesh-independent INRs (Tong et al., 2024).

Noise robustness is a major differentiator. In the comparative EIT study, conditional score-based diffusion generalizes best under high noise, while conditional normalizing flows are best at low noise (Wang et al., 2023). CDEIT remains strong at 40 dB noise but loses ground at 30 dB to physics-based refinement baselines (Shi et al., 2024). CDMVC explicitly inserts a voltage-consistency constraint during sampling to improve robustness and fidelity (Shi et al., 10 Jan 2025). RDPS adds explicit regularization to the diffusion prior precisely because implicit learned priors alone can be insufficient for severely ill-posed nonlinear inverse problems (Alberti et al., 19 May 2026).

Feature choice is another recurring issue. In perovskite EIS inversion, equivalent-circuit features outperform minimalist peak features because subtle high-frequency structures are required for some targets (Nabil et al., 10 Nov 2025). In battery and electrochemical systems, bounded-diffusion or diffusion-time-distribution models recover physically meaningful parameters only if geometry, size distribution, or dielectric layers are modeled explicitly [(Hallemans et al., 2024); (Song et al., 2017); (Khazimullin et al., 2018); (Song et al., 2012)]. This suggests that diffusion-based impedance learning is highly sensitive to the representation in which the inversion is posed: raw signals, engineered features, mesh-native states, or latent images can all be effective, but only when aligned with the underlying physics.

A further recurring theme is bias or operating-point dependence. Perovskite work shows that open circuit and short circuit expose different parameter classes (Nabil et al., 10 Nov 2025). Battery-model fitting requires simultaneous multi-SOC data (Hallemans et al., 2024). EIT reconstruction performance depends on mask patterns, noise level, and whether completion, posterior sampling, or direct conditional generation is used (Chen et al., 8 Feb 2026, Alberti et al., 19 May 2026). In robotics, force-rich, repetitive tasks particularly benefit from diffusion policies, while simpler short-horizon insertion may not require them to the same degree (Aburub et al., 2024).

7. Outlook and synthesis

Diffusion-Based Impedance Learning is converging toward an overview of three ingredients: a physically faithful forward model, a learned generative prior or denoiser, and an inversion mechanism that respects the structure of the measurement space. In drift–diffusion perovskite inversion, the forward model is already a first-principles simulator and the learning module mainly accelerates inverse mapping (Nabil et al., 10 Nov 2025). In EIT, learned priors compensate for the extreme ill-posedness of the conductivity inverse problem, but the best-performing methods increasingly reincorporate physics during sampling through voltage consistency, likelihood gradients, or finite-element posterior guidance (Shi et al., 10 Jan 2025, Alberti et al., 19 May 2026). In electrochemistry, bounded diffusion and diffusion-time distributions show that even before machine learning, physically grounded diffusion operators were indispensable for interpretable impedance inversion [(Song et al., 2017); (Song et al., 2012); (Khazimullin et al., 2018)]. In robotics, diffusion models provide a tractable way to represent multimodal, contact-conditioned action or compliance distributions that are difficult to model with fixed-depth deterministic policies (Aburub et al., 2024, Geiger et al., 24 Sep 2025, Okada et al., 2024).

This suggests that the most stable future formulations will likely be hybrid rather than purely data-driven. One explicit proposal along these lines appears in the perovskite study, which notes that probabilistic diffusion generative models could be combined with drift–diffusion-generated training sets to yield uncertainty-aware inversion under noise and domain shift (Nabil et al., 10 Nov 2025). Related EIT work already demonstrates analogous hybrids through diffusion posteriors guided by PDE likelihoods or learned forward surrogates (Shi et al., 10 Jan 2025, Alberti et al., 19 May 2026). The broader pattern is that impedance data, whether electrical or mechanical, become learnable when diffusion supplies the right latent geometry: either as the governing physics of transport, or as the generative mechanism that regularizes inverse inference.