Variational Autoencoder Inverse Mapper (VAIM)

• VAIM is a VAE-based inverse mapper that learns low-dimensional latent representations to enable both inverse design and forward prediction across diverse data domains.
• It employs dual VAEs to model device images and response curves independently, using polynomial regressors to establish deterministic latent–latent mappings.
• Empirical results demonstrate high prediction accuracy and robust noise handling, highlighting VAIM's effectiveness in semiconductor modeling, lattice QCD, and other inverse problems.

Searching arXiv for recent and foundational papers on VAIM and closely related inverse-mapping VAE frameworks. The Variational Autoencoder Inverse Mapper (VAIM) is a class of VAE-based inverse-problem frameworks in which low-dimensional latent representations are learned for one or more data domains and then used to construct inverse and forward mappings between them. In the concrete realization introduced for semiconductor device modeling, VAIM is implemented by learning one manifold for device structure images and another for $I$–$V$ characteristics, then connecting those manifolds through latent–latent mappings so that the same system supports both inverse design and forward prediction (Lu et al., 2023). In later usage, the term VAIM is also applied to inverse mappings from lattice-QCD observables to gluon parton distribution functions, from DVCS observables to Compton form factors, and to Bayesian inverse problems more broadly, indicating a methodological family rather than a single architecture (Kriesten et al., 23 Jul 2025).

1. Definition and conceptual scope

In the device-physics formulation, the method in (Lu et al., 2023) is described as “essentially a concrete realization of a ‘Variational Autoencoder Inverse Mapper (VAIM)’ between device structure images and I–V characteristics.” Its defining feature is an invertible mapping, up to the usual VAE stochasticity, between a structural domain and a response domain, obtained by learning low-dimensional manifolds for both and connecting them through a learned latent–latent mapping (Lu et al., 2023).

In that formulation, the two VAEs define two manifolds, one for structure images and one for curves, and the polynomial regressors define an approximate bijection between points on these manifolds. This makes VAIM simultaneously an inverse mapper and a forward mapper: structure $\rightarrow$ latent $\rightarrow$ latent $\rightarrow$ response for compact modeling, and response $\rightarrow$ latent $\rightarrow$ latent $\rightarrow$ structure for inverse design (Lu et al., 2023).

Subsequent work uses the same label for related inverse-problem settings. In lattice QCD, VAIM is a deep generative model for extracting the gluon PDF from reduced pseudo-Ioffe-time distributions, where observables and physics parameters are linked through a structured latent representation (Kriesten et al., 23 Jul 2025). In DVCS phenomenology, VAIM and C-VAIM are used to extract Compton Form Factors from cross sections while preserving multiple admissible solutions and exploiting cross-kinematic correlations (Hossen et al., 2024). This suggests that “VAIM” functions as a general designation for VAE-based inverse mappers in which uncertainty, latent structure, and inverse ambiguity are treated explicitly.

2. Device-image to $I$–$V$ VAIM architecture

The device-image VAIM in (Lu et al., 2023) consists of three parts: an image VAE, a curve VAE, and latent–latent mapping via polynomial regressors. The full framework consists of an image-VAE whose input is a device cross-section image as a monochrome PNG of $V$0 pixels, and a curve-VAE whose input is an $V$1–$V$2 curve discretized to 51 points for $V$3 from 0 to 1.4 V at $V$4 V (Lu et al., 2023).

The image-VAE encoder is a stack of Conv2D layers followed by Flatten and Dense layers, ending in latent mean and log-variance and sampling. Its latent dimensionality is 30. The decoder uses Dense, Reshape, and Conv2D Transpose layers back to $V$5. Hidden activations are ReLU and the final image output layer uses sigmoid. The image reconstruction loss is binary cross-entropy on the pixels (Lu et al., 2023).

The curve-VAE encoder is built from fully connected layers ending in latent mean and log-variance. Its latent dimensionality is lower than the 51-point curve, and in the figure is shown as 5. The decoder maps back to 51 outputs, with ReLU in hidden layers and a linear final output layer. The curve reconstruction loss is mean squared error on the curve values (Lu et al., 2023).

The latent code of the image VAE is denoted $V$6, and the latent code of the curve VAE is denoted $V$7. A set of third-order polynomial regressors maps between the two latent spaces in both directions: $V$8 These mapping networks are trained after both VAEs are trained, using paired image and $V$9–$\rightarrow$0 data and their corresponding latent codes (Lu et al., 2023).

The resulting forward mapping is: $\rightarrow$1 while the inverse mapping is: $\rightarrow$2 These compositions are the core operational definition of VAIM in the paper (Lu et al., 2023).

3. Variational formulation and manifold coupling

Each VAE in (Lu et al., 2023) follows the standard variational autoencoder framework, maximizing an ELBO for generic data $\rightarrow$3, latent $\rightarrow$4, encoder $\rightarrow$5, and decoder $\rightarrow$6: $\rightarrow$7 with standard Gaussian prior $\rightarrow$8 (Lu et al., 2023).

For the image-VAE, the reconstruction term is BCE and the KL term is the divergence between

$\rightarrow$9

and the standard normal prior. For the curve-VAE, the reconstruction term is MSE on the 51 curve values and the KL term is defined analogously for

$\rightarrow$0

The total losses are reconstruction plus KL in each domain (Lu et al., 2023).

The coupling between the two VAEs is not a shared latent space in the strict probabilistic sense. Instead, it is done via deterministic regressors between the two latent spaces after the VAEs are trained: $\rightarrow$1 with least-squares fitting of $\rightarrow$2 and $\rightarrow$3 over paired latent codes (Lu et al., 2023). This separation between manifold learning and manifold coupling is one of the clearest distinguishing features of the original VAIM realization.

A broader implication, supported by later VAIM variants, is that the inverse-mapping function need not always take the same architectural form. In the lattice-QCD VAIM, the latent manifold is partitioned into a deterministic observable part and a stochastic generative part, and the decoder maps observables plus latent samples back to PDF parameters (Kriesten et al., 23 Jul 2025). In Bayesian inverse-problem formulations such as UQ-VAE and eUQ-VAE, the encoder itself becomes the inverse mapper $\rightarrow$4, while the decoder is the known or learned forward map (Goh et al., 2019, Tonini et al., 18 Feb 2025). This suggests that the common element across VAIM formulations is not a single canonical network topology, but the use of VAE-style latent probabilistic structure to regularize and operationalize inverse mappings.

4. Inverse design, forward prediction, and compact modeling

In (Lu et al., 2023), inverse design means that given a desired $\rightarrow$5–$\rightarrow$6 curve, the model produces a structure image whose simulated $\rightarrow$7–$\rightarrow$8 matches that target curve, in particular its $\rightarrow$9 and $\rightarrow$0. The inference procedure is explicit: encode the desired curve into curve latent space, map to image latent space through the polynomial regressor, and decode to a structure image (Lu et al., 2023).

For noisy “hand-drawn” curves, the paper reports that it is beneficial to pass the curve twice through the curve-VAE before the inverse path, and to pass the final images three times through the image-VAE after the inverse path, as a denoising and manifold-consistency step (Lu et al., 2023). Validation is then performed by converting the generated structure image to a TCAD structure, running full TCAD $\rightarrow$1–$\rightarrow$2 simulations, extracting $\rightarrow$3 and $\rightarrow$4, and comparing these to the desired curves.

For forward prediction, the same framework acts as a compact model that uses the device image itself as the “parameter vector.” The path is image $\rightarrow$5 image latent $\rightarrow$6 curve latent $\rightarrow$7 predicted $\rightarrow$8–$\rightarrow$9 curve (Lu et al., 2023). For noisy hand-drawn images, the image is first passed through the image-VAE once to denoise and project it onto the learned manifold.

The reported results on hand-drawn images are $\rightarrow$0 for $\rightarrow$1 prediction and $\rightarrow$2 for $\rightarrow$3 prediction. For inverse design using noisy curves, the reported values are $\rightarrow$4 for $\rightarrow$5 and $\rightarrow$6 for $\rightarrow$7 (Lu et al., 2023). These figures are central to the empirical characterization of VAIM in its original device-physics setting.

A concise summary of the principal roles in the original framework is as follows.

Function	Input	Output
Forward prediction	Device structure image	Predicted $\rightarrow$8–$\rightarrow$9 curve
Inverse design	Desired $\rightarrow$0–$\rightarrow$1 curve	Predicted device image
Compact modeling view	Device image treated as parameters	Entire $\rightarrow$2–$\rightarrow$3 curve

The same duality between forward and inverse use appears in later VAIMs, although with different observables and targets. In DVCS applications, the forward map sends CFFs to cross sections while the inverse mapper recovers distributions over CFFs from observed cross sections (Hossen et al., 2024, Almaeen et al., 2024). In lattice QCD, the training is forward in the sense that PDF parameters generate RpITDs through the matching relation, but inference is performed backward by sampling latent states conditioned on observables to obtain distributions of gluon PDFs (Kriesten et al., 23 Jul 2025).

5. Noise robustness, irrelevant variables, and latent structure

The original device-image VAIM is explicitly framed as a manifold-learning system. The image-VAE compresses $\rightarrow$4 images into a 30-dimensional latent space, while the curve-VAE compresses 51-point curves into a lower-dimensional curve space. The paper motivates this by noting that only five physical parameters vary: $\rightarrow$5, $\rightarrow$6, $\rightarrow$7, $\rightarrow$8, and $\rightarrow$9 (Lu et al., 2023).

The framework uses five independent variables with different strengths: strong variables $\rightarrow$0 and $\rightarrow$1, medium variable $\rightarrow$2, and weak or irrelevant variables $\rightarrow$3 and $\rightarrow$4. In inverse design, the machine is described as having learned to assume a constant $\rightarrow$5 and $\rightarrow$6 for all devices, consistent with the fact that these parameters barely affect the $\rightarrow$7–$\rightarrow$8 curves (Lu et al., 2023). This is presented as evidence that the latent manifold disregards directions with little influence on the response.

The paper also notes that 8000 training samples are sufficient for five parameters, corresponding to roughly $\rightarrow$9 effective resolutions per parameter, consistent with earlier manifold-learning work (Lu et al., 2023). Since all five physical variables are only implicit in the images and curves and are not fed explicitly, the framework learns the parametric effects directly from raw pixels and curve points.

Noise handling is built into both the VAE regularization and the practical inference protocol. For inverse design, 20 noisy “hand-drawn” curves are used, with noise omitted at $I$0 V, 0.028 V, 1.372 V, and 1.4 V so that $I$1 and $I$2 remain unchanged (Lu et al., 2023). For forward prediction, 10 hand-drawn variants of unseen TCAD structures are created by changing greyscale intensities, removing doping gradients, and making boundaries unsmooth while preserving the five geometric parameters.

The system’s robustness to noise and ambiguity is a recurring motif across later VAIM literature. In the DVCS applications, VAIM is used precisely because a single unpolarized cross section can correspond to infinitely many CFF solutions; the latent variable is used to preserve and sample that non-uniqueness (Hossen et al., 2024, Almaeen et al., 2024). In Bayesian inverse-problem formulations, VAIM-like architectures are used to produce full approximate posteriors rather than point estimates, making uncertainty quantification a first-class output of the inverse mapper (Goh et al., 2019, Tonini et al., 18 Feb 2025). This suggests that VAIM is most naturally understood not as a deterministic inverse regressor, but as a latent-manifold inverse framework suited to ill-posed mappings.

6. Implementations, variants, and methodological lineage

The original device-image VAIM is one instance within a broader methodological lineage of VAE-based inverse mappers. Several later papers clarify different architectural choices that remain compatible with the VAIM concept.

In the lattice-QCD-informed machine-learning formulation, the VAIM solves the inverse problem of extracting the gluon PDF $I$3 from 35 lattice RpITD points. The parameter space is

$I$4

the observable space is

$I$5

and the model partitions its latent state into a deterministic observable part and a 256-dimensional stochastic generative latent variable $I$6 (Kriesten et al., 23 Jul 2025). The decoder takes $I$7 and reconstructs PDF parameters, allowing an ensemble of PDFs to be generated from an ensemble of RpITDs.

In the DVCS CFF-extraction setting, VAIM is formulated as a forward mapper $I$8 and inverse mapper $I$9, with the latent variable capturing the information lost in the many-to-one map from CFFs to cross sections. C-VAIM adds conditioning on kinematic variables $V$0, effectively learning correlations among different kinematic bins and narrowing the extracted CFF distributions (Hossen et al., 2024). The related VAIM-CFF formulation places special emphasis on using the latent space to analyze “missing physics information” in the forward map (Almaeen et al., 2024).

In Bayesian inverse problems, UQ-VAE reframes the latent variable as the physical parameter itself, with the encoder learning $V$1 and the decoder given by the known forward map $V$2 or a learned surrogate $V$3 (Goh et al., 2019). The eUQ-VAE variant derives a new loss from an upper bound on a Jensen–Shannon-divergence-based objective and proves a posterior-consistency result in the affine linear-Gaussian case (Tonini et al., 18 Feb 2025). These formulations move away from the dual-VAE-plus-latent-mapper design of (Lu et al., 2023), but preserve the core VAIM idea of using VAE machinery to learn an inverse map to a distribution over underlying causes.

Additional variants extend the concept further. Half-VAE removes the encoder entirely and treats the latent variables as trainable parameters, thereby bypassing explicit inverse mapping; the paper explicitly interprets this as a limiting case of a VAE-based inverse mapper (Wei et al., 2024). RegAE in hydrologic inverse analysis uses a trained VAE decoder as a learned regularizing manifold so that inversion is performed in latent space and then decoded back to a high-dimensional conductivity field (O'Malley et al., 2019). In geophysical imaging, VAE-based latent inversion is analyzed as a tradeoff between pattern fidelity and inversion feasibility, with the decoder acting as the learned inverse parameterization of the field (Lopez-Alvis et al., 2020). Joint Posterior Maximization with Autoencoding Prior similarly uses a VAE as a reusable prior for ill-posed imaging inverse problems, optimizing jointly over image space and latent space rather than learning a direct inverse network (González et al., 2021).

A distinct but related uncertainty-aware extension appears in the Copula-based VAE for damage identification in floating offshore wind turbines, where the encoder approximates the inverse operator and the decoder approximates the forward one, while the approximate posterior over physical damage variables is modeled by either Gaussian mixtures or a Gaussian copula (Fernandez-Navamuel et al., 2 Oct 2025). The paper’s emphasis on flexible posterior modeling is consistent with the broader VAIM objective of representing multimodal or correlated inverse uncertainty.

7. Applications, significance, and open directions

The original device-image VAIM is positioned as useful for compact modeling of novel devices when only device cross-sectional images and electrical characteristics are available, including “novel emerging memory” (Lu et al., 2023). The authors further state that the framework can be “quickly deployed on any new device and measurement” because it does not require explicit parameter extraction or domain knowledge about which geometric features matter (Lu et al., 2023). A plausible implication is that VAIM is especially attractive in regimes where geometry is observable but the most appropriate hand-engineered descriptors are not yet settled.

The same device-agnostic claim is echoed by the range of later VAIM applications. The lattice-QCD VAIM is presented as a bridge between lattice calculations and phenomenological PDF extractions within a unified analysis framework (Kriesten et al., 23 Jul 2025). DVCS VAIMs are presented as a first step toward a broader GPD-extraction pipeline (Hossen et al., 2024, Almaeen et al., 2024). Hydrologic and imaging inverse formulations show that VAE-based learned priors can regularize otherwise under-constrained inverse problems by reducing optimization to a low-dimensional latent space (O'Malley et al., 2019, Lopez-Alvis et al., 2020, González et al., 2021).

Several recurring limitations also emerge. Parametrization dependence remains important in settings where the target object is represented by a restricted functional family, as in the gluon-PDF parameterization $V$4 (Kriesten et al., 23 Jul 2025). Gaussian approximate posteriors can be too restrictive in Bayesian inverse settings, motivating richer families such as copulas or Gaussian mixtures (Goh et al., 2019, Fernandez-Navamuel et al., 2 Oct 2025). In device-image VAIM, some latent dimensions are intentionally allowed to collapse weak physical factors, which is desirable for robustness but also means the inverse image need not reproduce all nominal device parameters uniquely (Lu et al., 2023).

A common misconception is that VAIM denotes a single standardized architecture. The literature does not support that interpretation. In (Lu et al., 2023), VAIM is realized by stacked VAEs plus third-order polynomial latent regressors. In (Kriesten et al., 23 Jul 2025), it is a conditional generative model with an observable latent block and a 256-dimensional stochastic block. In (Goh et al., 2019) and (Tonini et al., 18 Feb 2025), the encoder itself is the inverse mapper from observations to an approximate posterior over parameters. In (Wei et al., 2024), explicit inverse mapping is bypassed altogether. The unifying idea is instead the use of VAE-style latent probabilistic structure to turn ill-posed inverse mappings into tractable forward–inverse systems.

In this sense, VAIM can be understood as a family of inverse-learning strategies built around three recurrent commitments: low-dimensional latent structure, probabilistic regularization, and explicit accommodation of inverse ambiguity. The device-image framework of (Lu et al., 2023) remains the clearest concrete instantiation of the term, but later work shows that its scope extends well beyond semiconductor compact modeling into QCD, hadronic phenomenology, engineering diagnostics, hydrology, and Bayesian scientific inversion.