Ψ-Vortex: Structure-Regularized Recurrent Learning for Latent Thermal-Coupling Inference and Verilog-A Compact Modeling of 3-D Neuromorphic Devices

Today, my latest work, “Ψ-Vortex: Structure-Regularized Recurrent Learning for Latent Thermal-Coupling Inference and Verilog-A Compact Modeling of 3-D Neuromorphic Devices,” has been published in IEEE Access (IF: 4.2).

Ψ-Vortex is a structure-regularized recurrent learning framework that connects history-dependent device modeling, latent-state inference, structural compression, and executable Verilog-A compact models. At the technical level, the work asks how recurrent learning can be made sufficiently structured, reproducible, and compact to move beyond neural-network fitting toward circuit simulation. The deeper question is more specific: can a learning system recover latent physical organization from electrical trajectories, determine the regime in which that inference remains reliable, and translate the resulting representation into a compact model that executes consistently in a circuit simulator?

That question emerged gradually from the previous stages of my work. The immediate starting point was Ψ-xLSTM, where I moved from predominantly pointwise compact modeling toward recurrent architectures capable of representing fast, history-dependent device dynamics. Electronic devices do not exist only in steady state. They switch, relax, drift, accumulate history, and can respond differently to the same instantaneous input depending on what happened before it. A model that sees only the present voltage and current can therefore reproduce a curve while still failing to represent the evolving internal state that generated it.

Once Ψ-xLSTM established that recurrent modeling direction, several deeper problems became visible. A high-capacity recurrent teacher can capture complicated temporal behavior, but it is expensive compared with the compact equations ultimately required by a circuit simulator. Earlier Ψ-family workflows also retained expert decisions about clustering, compact representation, and architecture selection, so a substantial part of the pipeline still depended on manual inspection. More fundamentally, good recurrent prediction alone does not demonstrate that temporal state is being used in a scientifically meaningful way. A sufficiently flexible model can fit observed outputs without necessarily learning the historical dependence we are trying to identify. Ψ-Vortex therefore became less about constructing another recurrent network and more about developing a workflow in which recurrence, latent inference, structure selection, compression, and circuit execution could each be tested separately.

From physical priors to testable evidence

One of the first ideas I explored was physics-aware initialization. If some physical organization is plausible before training begins, forcing the optimizer to start from a completely unstructured parameter state may be inefficient. Ψ-Vortex therefore allows symmetry-related information to influence the initial state of eligible weight matrices while preserving orthogonal initialization for recurrent matrices. The purpose is not to hard-code the solution or permanently constrain paired weights, but to test whether a physically informed starting point can improve optimization under particular conditions.

The experiments produced a more nuanced result than a universal speedup. On the declared printed-sequence task, physics-aware initialization reached the shared target in an average of 6.8 epochs, compared with 462 epochs for random initialization, corresponding to a recorded 116.6-fold acceleration. On the thermal-coupling task, however, random initialization produced better test fidelity than the physical-retention initialization. That result changed the interpretation of the method: physics-aware initialization can accelerate optimization dramatically in one regime without guaranteeing better transfer or final fidelity in another. The correct conclusion is therefore that initialization is task-dependent and should be evaluated rather than assumed beneficial.

The same principle guided the automatic symmetry detector. Instead of forcing the framework to declare odd or even symmetry, the detector can return odd, even, or none, depending on what the data support. In the final experiments, it selected none on every tested seed, which means it behaved like the identity configuration rather than imposing a symmetry that the evidence did not justify. I see this as a desirable property of scientific automation.

The structure-selection problem led to another component of Ψ-Vortex. Classical Bayesian Information Criterion balances model fit against complexity, but its discrete parameter-count formulation does not translate naturally into gradient-based neural training. Ψ-Vortex therefore uses a differentiable BIC-inspired structural objective, rather than claiming to implement classical BIC directly. The student objective combines supervised target fidelity, recurrent relation-aware distillation, and structural regularization, while validation data are used to select compact representations before final testing and export. The framework also evaluates a bounded set of 36 candidate architectures. Across the five declared seeds, five different configurations were selected, so the result does not support a claim that one architecture is universally optimal. Instead, Ψ-Vortex provides a reproducible procedure for selecting among declared candidates without using the test set to make that choice. Search bounds, candidate clusters, bandwidths, and training budgets remain user-defined.

Why three-dimensional neuromorphic hardware became the harder test

The most challenging experiment came from three-dimensional neuromorphic hardware. Printed vertical conductors have already demonstrated signal transmission through structures containing 1000 layers, which suggests a path toward extremely dense printed electronic architectures. Such integration also introduces interactions that are difficult to observe directly. Every switching event can generate heat, and in a multilayer system that heat can propagate through substrates, interfaces, conductors, and neighboring structures. A device may therefore display an electrical response that depends partly on an internal thermal history that is not directly available at its terminals.

The 1000-layer architecture provides the motivation for this problem, but it is not the system evaluated by Ψ-Vortex. The latent-coupling benchmark itself is deliberately smaller and more controlled. I constructed a calibrated synthetic two-layer driver-victim configuration as the minimal pairwise unit of the broader stacking problem. A driver memristor occupies one layer, while a victim device sits on the adjacent layer across a 100 micrometer paper substrate. The victim is held at a constant low read voltage below its reported switching threshold, which reduces the possibility that its response can simply be explained by direct electrical switching. The model receives electrical trajectories, while the effective thermal-coupling coefficient is not supplied as a training label. The question is whether a stateful recurrent model can encode the slowly evolving influence of the driver and whether that effective coupling can then be recovered from disjoint held-out trajectories.

This is where the chronology controls become necessary. The normal Ψ-Vortex configuration preserves temporal order and recurrent state across contiguous portions of each source trajectory. I then deliberately broke different parts of that temporal contract by resetting recurrent state, shuffling temporal order, or removing memory entirely. At the nominal coupling α=0.08, chronological Ψ-Vortex passed all three seeds, while state reset, shuffled order, and pointwise modeling passed none of their nine combined runs. The result shows that successful recovery depends on temporal ordering and state continuity rather than only on instantaneous input-output fitting.

The conventional LSTM baseline produced one of the clearest conclusions in the study. A chronological vanilla LSTM passed 14 of the 15 conditions across the broader coupling sweep. This means the evidence does not support a claim that one particular recurrent cell is uniquely capable of recovering the interaction. Instead, it supports the broader conclusion that chronological stateful recurrence is the enabling model-class property. The ability to integrate device history over time is the central requirement; the result is not unique to the Ψ-Vortex recurrent cell.

Recovering latent coupling and defining the boundary of the claim

At the nominal coupling α=0.08, Ψ-Vortex recovered α^=0.08738 with a held-out R2=0.981, and all three declared seeds passed the acceptance criterion. The weaker tested coupling, α=0.05, also passed every seed, with a mean recovered value of approximately 0.0523 and a mean error of 4.6 percent. The behavior changed at stronger coupling. At α=0.10, the mean recovery error rose to 36.0 percent and only one of three seeds passed; at 0.15 the mean error reached 56.5 percent, again with one passing seed; and at 0.20 the mean error was 58.6 percent, with no seed passing. The evidence therefore supports a bounded conclusion: reproducible coupling recovery was obtained for the weaker tested pairwise regime represented by α=0.05 and 0.08, but not across the full sweep.

I prefer that bounded result to a broader claim based only on the nominal case because it tells us where the current method stops being reliable. A modeling framework should provide evidence not only for successful cases, but also for the conditions under which its interpretation can no longer be defended. In Ψ-Vortex, the higher-coupling failures define that boundary directly rather than leaving it implicit.

A separate limitation concerns the interpretation of the thermal experiment itself. This is a proof-of-concept on calibrated synthetic data, not an experimental discovery of thermal coupling inside a fabricated three-dimensional memristor stack. The benchmark generator contains a known thermal-electrical relationship, and the post-hoc coefficient recovery benefits from the functional-form match between that generator and the estimation procedure. The recovered α should therefore be interpreted as an effective coupling descriptor within the controlled benchmark, not as a universal thermal material constant. Fabricated multilayer devices, distributed heat sources, stronger nonlinear interactions, stochastic variability, larger arrays, and more complicated interface physics remain future tests. The published paper explicitly treats fabricated-stack thermal discovery and reliable multilayer recovery as open problems.

Curve fitting and latent-state inference are different tasks

I also wanted to determine whether the apparent benefit of recurrence could simply be explained by better current-voltage fitting. Ψ-Vortex was therefore compared with a conventional MLP, a physics-informed Kolmogorov-Arnold network (PIKAN), and SINDy. On measured-device fitting, the static MLP and PIKAN remained highly competitive, while SINDy showed greater variation across devices. This separates two tasks that are often treated as though they were equivalent: reproducing an observed waveform and recovering a hidden history-dependent interaction.

Across the synthetic coupling sweep, the MLP, PIKAN, and SINDy baselines each passed none of their 15 latent-recovery conditions, while the chronological vanilla LSTM passed 14 of 15. Static models can therefore fit simple measured switching trajectories very well and still fail a task that requires accumulated temporal state. The distinction is not between a good curve-fitting model and a bad one. It is between a pointwise mapping and a model capable of representing the temporal information required by the hidden process.

The independent measured-device experiments provide a complementary test of the recurrent compact-modeling pipeline. On three measured graphene-oxide/polyelectrolyte memristive devices, Ψ-Vortex obtained held-out seed-mean correlations of approximately 0.9973, 0.9999, and 0.9980. Cross-device cell-mean correlation remained at least 0.974, and cross-rate correlation remained at least 0.976. These experiments show that the recurrent pipeline can reproduce measured switching trajectories and transfer across device and excitation conditions. They do not validate thermal-coupling discovery in fabricated devices because the measured dataset contains no corresponding thermal ground truth. The measured experiments therefore support recurrent compact-model fidelity, while the calibrated synthetic benchmark supports the more specific latent-coupling analysis.

From a recurrent teacher to an executable compact model

The next question was whether the recurrent representation could be reduced enough to become practical for circuit simulation. A circuit simulator does not need the largest neural model that can be trained; it needs a representation that preserves the required behavior while remaining compact and executable. In the canonical synthetic case, the Ψ-Vortex workflow reduces the teacher from 26,273 expanded parameters to 273 expanded parameters, corresponding to a 98.96 percent reduction. The clustered representation contains 16 centroid values, but those values should not be interpreted as 16 physical circuit components. A centroid is a shared learned parameter value in the compact representation, not evidence for a one-to-one set of resistors, capacitors, thermal elements, or microscopic mechanisms. For that reason, the paper reports centroid count, expanded parameter count, expanded parameter bytes, and serialized artifact size separately rather than treating them as equivalent measures of compactness.

Compression alone is not enough, however, because a compact model that works only inside PyTorch is still separated from conventional electronic-design workflows. Validation-selected recurrent students were therefore exported to Verilog-A, compiled with OpenVAF-reloaded, and executed through OSDI in ngspice. Five test stimuli passed the declared simulator-consistency criterion, with the worst absolute discrepancy relative to the corresponding PyTorch execution equal to approximately 5.46×10^{-5}, below the predefined 10^{-3} tolerance. This establishes equation-level circuit execution. It does not establish PDK qualification, fabricated-stack validation, or process-corner robustness, but it closes the demonstrated path from ordered electrical trajectories through recurrent inference and structural compression to an executable compact model inside a circuit simulator.

That full path is what I see as the central contribution of Ψ-Vortex. The framework combines chronological recurrent learning, controlled latent-coupling inference, validation-guided structural regularization, compact recurrent representation, and Verilog-A execution, while the experiments also expose the boundaries of each claim. Physics-aware initialization can accelerate one task dramatically while producing worse fidelity on another. Automatic symmetry detection can abstain when the tested symmetry is unsupported. Static models can reproduce measured curves with high fidelity while failing a history-dependent latent-inference task. Stateful recurrence supports the latent-recovery problem, but the result is not unique to one recurrent cell. Coupling recovery succeeds within a bounded pairwise regime rather than across every tested condition. Verilog-A execution verifies simulator consistency, but it is not equivalent to fabricated-device or PDK validation. Together, these results produce a framework whose limitations are as explicit as its successful cases.

From Ψ-NN to Ψ-Vortex

The intellectual path to Ψ-Vortex began before Ψ-HDL and Ψ-xLSTM. A major turning point for me was discovering Ψ-NN, introduced by Liu et al. here. Their work showed that physics-informed learning could be approached not only as a method for approximating solutions, but also as a way to extract and reconstruct structure from what a neural model had learned. That idea changed the direction in which I began thinking about compact modeling. Without encountering Ψ-NN, I would probably not have developed the subsequent Ψ-family in its current form. I even visited China a few months after that. I had such a great time, that my name even appeared on some newspapers.

Ψ-HDL extended that idea toward compact device modeling and automated Verilog-A generation. Ψ-xLSTM then moved the methodology toward recurrent modeling of fast and history-dependent device dynamics. Ψ-Vortex brings chronological recurrence, validation-guided structural compression, latent-coupling inference, and compiled circuit execution into a common workflow.

Ψ-Vortex is my current answer to the latest part of that sequence, but it also makes the next stage much clearer. The thermal experiment now needs to move beyond a calibrated synthetic pairwise benchmark toward fabricated multilayer devices, larger device populations, distributed interactions, fabrication variability, incomplete measurements, and more demanding circuit-level operating conditions. That is where assumptions that behave cleanly in a controlled benchmark will encounter the irregularity of physical hardware.

When I started developing Ψ-HDL, I was primarily trying to automate compact-model generation. I did not expect that a few iterations later I would be using recurrent models to infer an effective latent thermal interaction from electrical trajectories and then translating the resulting recurrent representation into an executable circuit model. That progression reflects how this research direction has developed for me: solving one technical bottleneck has repeatedly exposed a more difficult question behind it, and each new Ψ-family framework has been an attempt to address the next limitation rather than simply extend the previous model.