Earlier this month, I wrote about submitting PSI-MEM, my first MSCA Postdoctoral Fellowship proposal, and about the scientific question at its centre. Can structural regularities in printed-memristor dynamics remain reproducible despite cycle-to-cycle, device-to-device, and fabrication-batch variability? Since then, I have continued thinking about the broader idea behind that question. Not specifically about the proposal itself, but about what Physics Structure-Informed (PSI) modeling could mean as a more general research direction for emerging electronics.
The Ψ-family has already explored several parts of this problem. Ψ-HDL connected structure discovery with compact-model and Verilog-A generation. Ψ-xLSTM extended the approach toward recurrent and multi-timescale dynamics. Ψ-Vortex moved further toward chronology-aware inference, structural regularization, validation, compression, and simulator-executable models. So the question for me is no longer simply how to build another member of this family. What interests me now is the level above the individual framework. If a model discovers compact organization in an electronic device, what would make us trust that organization? Does it remain when an experiment is repeated? Does it appear in another nominally similar device? Does it survive changes in operating conditions or fabrication?
That leads to a broader way of thinking about device modeling. Perhaps the goal should not only be to predict device behaviour accurately, but also to determine which learned structures remain supported when the physical system itself varies.
Why emerging electronics are difficult to model
Emerging electronics are expanding the design space of electronic systems through new materials, switching mechanisms, form factors, and manufacturing routes. Printed and flexible electronics, memristive devices, neuromorphic and flexible sensor platforms, oxide and organic electronics, can provide local memory, nonlinear response, mechanical flexibility, additive fabrication, and new possibilities for edge intelligence. The same characteristics that make these technologies attractive also make them difficult to model because their electrical behaviour can depend on previous excitation and multiple characteristic time scales, internal state variables that are not directly observable, interfaces, morphology, material composition, processing conditions, and fabrication variability.
For printed memristive devices in particular, the measured response may change between repeated cycles of the same device, between independently fabricated devices, and between fabrication batches. Circuit and system design therefore requires more than an accurate reproduction of one measured current-voltage curve. A useful compact model must describe the relevant device dynamics efficiently enough for repeated circuit simulation while remaining meaningful beyond the particular measurements from which it was constructed. This becomes especially difficult when the dominant physical mechanisms are only partially understood and nominally similar devices do not behave identically.
What existing modeling approaches already provide
Conventional compact modeling begins with a mathematical description of the device. Physics-based and phenomenological models can be efficient, interpretable, and highly practical when the dominant mechanism and a suitable set of state equations are sufficiently understood. The difficulty appears when the appropriate representation is uncertain because the equations, internal variables, switching laws, and functional dependencies generally have to be selected before parameter extraction begins. For emerging devices with incomplete physical knowledge or substantial variability, selecting the model form can therefore become part of the scientific problem itself.
Data-driven modeling approaches begin from the opposite direction. Neural networks and other machine-learning methods can learn complicated device responses directly from measurements, and previous work has shown that learned representations can be translated into Verilog-A and integrated into circuit simulation. Messaris et al. demonstrated a data-driven Verilog-A ReRAM model, while Hutchins et al. presented a more general machine-learning compact-modeling workflow for multi-state devices. Physics-informed neural networks narrow the gap between physical and data-driven modeling by incorporating governing equations, device assumptions, or physical restrictions into learning. Physics-informed recurrent models additionally address history-dependent RRAM dynamics, while PINN-based memristor compact models have been implemented in Verilog-A.
Sparse and symbolic discovery methods offer another route by attempting to recover explicit dynamical equations from observations. Physics-enhanced sparse identification can incorporate prior physical knowledge, but the result still depends on the available observables, coordinate representation, candidate function space, and treatment of hidden states. Hard nonlinearities and unobserved dynamics remain particularly difficult cases. These approaches should not be viewed as obsolete alternatives. Each addresses a different part of the modeling problem. The question that increasingly interests me is whether the internal organization of the model itself can become something that is discovered, challenged, and potentially rejected.
From physics-informed prediction to physics-informed structure
The conceptual starting point for this direction came from Ψ-NN, introduced by Liu and colleagues in 2025. Ψ-NN changed the usual role of a physics-informed neural network by separating physical regularization from parameter regularization through teacher-student distillation. A physics-informed teacher first learns the target problem under physical constraints. Its behaviour is then transferred to a student network, where parameter regularization and structure extraction can expose recurring relations. Those relations are subsequently embedded into a reconstructed structured network.
The important idea is deeper than parameter reduction. Physical information can become reflected in the organization of the learned representation itself. The Ψ-NN methodology explicitly treats structure discovery as an objective of the student representation, and its ablation analysis shows that distillation, structure extraction, and reconstruction contribute different pieces of the final structured model. This became the conceptual starting point for the Ψ-family methods I subsequently developed for electronic-device modeling.
Ψ-HDL connected discovered structure with circuit simulation
The first extension was Ψ-HDL. Its central question was whether discovered structure could become useful outside the neural network itself. Ψ-HDL combines physics-informed teacher models, knowledge distillation, hierarchical clustering, structure extraction, automatic Verilog-A generation, and circuit-level validation. In other words, the workflow attempts to transform learned internal organization into a compact representation that can participate in electronic-design workflows rather than remaining only inside a trained neural surrogate.
Across its case studies, Ψ-HDL showed that substantial structural compression could coexist with predictive capability and simulator-oriented deployment. The framework was tested on several types of problems, including memristive-device modeling, and the resulting behavioral descriptions were evaluated in circuit simulation. The conceptual step was therefore that structure discovery could become compact-model deployment. However, the feed-forward architectures used in Ψ-HDL are not sufficient for every emerging device, particularly when memory, hysteresis, relaxation, or fast temporal dynamics dominate the response.
Ψ-xLSTM extended structure discovery into time
Many emerging electronic devices cannot be represented adequately by treating individual operating points independently because their present behaviour depends on the sequence and timing of previous stimuli. This is particularly important when several characteristic time scales coexist. Ψ-xLSTM therefore moved PSI modeling into recurrent architectures by using an xLSTM-based physics-informed teacher and Recurrent Relation-Aware Distillation to preserve temporal behaviour during compression.
Structure discovery is then applied to the recurrent representation itself. Effective time scales can be extracted from exponential gating behaviour, while high-dimensional matrix memories can be approximated through lower-rank representations. These recurrent structures can subsequently be translated toward simulator-compatible behavioral models. This broadens the meaning of structure. It is no longer limited to static relations between neural parameters, but can also include temporal organization, effective characteristic time scales, recurrent dependencies, and compact modes within a higher-dimensional dynamical system.
Ψ-Vortex moved the focus toward structural validation
The next step was not simply to make the recurrent model larger. The more interesting problem was how to make structural inference less dependent on manually chosen settings and how to test whether the recovered organization was genuinely supported by temporal information. Ψ-Vortex extends the recurrent PSI direction through chronology-preserving learning, bounded validation-based architecture selection, symmetry detection, BIC-inspired structural regularization, latent-coupling inference, structural compression, and simulator-executable compact-model generation. The researcher still provides the data, target physical setup, assumptions, and search-space boundaries, so the goal is not fully automatic scientific discovery. It is instead to reduce several arbitrary structural decisions while keeping the scientific problem under researcher control.
Chronology is especially important because a system with memory can lose essential information when temporal order is destroyed. A model may still reproduce individual observations while failing to represent the underlying sequence-dependent process. Chronology-breaking controls therefore provide a way to test whether recurrent memory is genuinely contributing to the inferred relation. The compact-modeling side of Ψ-Vortex has also been evaluated on measured graphene-oxide and polyelectrolyte memristive devices. Its latent thermal-coupling result has a narrower interpretation because that inference was demonstrated using synthetic data calibrated to published device parameters, while validation of thermal coupling in fabricated multilayer systems remains future experimental work.
This distinction is important because structure is not automatically mechanism.
Compactness alone is not the endpoint
One attractive consequence of PSI learning is that complicated neural representations can sometimes be reduced dramatically. In Ψ-Vortex, validation-selected structural compression reduced a recurrent teacher with 26,273 parameters to an exported recurrent representation with 273 expanded parameters, corresponding to a 98.96 percent reduction. On measured graphene-oxide and polyelectrolyte memristive devices, held-out seed-mean correlations were approximately 0.997 to 1.000. Five generated Verilog-A models were also compiled and executed in ngspice, with the worst absolute discrepancy remaining approximately 5.46 × 10^-5 relative to the recurrent reference.
From single-device fitting to structure stability
What I increasingly see as the next scientific test for PSI modeling is not whether another architecture can improve predictive accuracy, but whether inferred structure remains reproducible under physical variability. Suppose a compact structural representation is identified from one device. The immediate question should not only be whether the same model can reproduce another attractive curve. The more important question is whether the inferred organization appears again after repeated cycling, under another operating condition, in another nominally similar device, or after another fabrication run.
This changes the problem from structure discovery to structure stability. A structure that remains reproducible across a population could support a shared compact representation. A structure that survives only within specific devices or operating regimes would require explicit validity boundaries and perhaps conditional digital twins. If no common structure survives, a distribution of device behaviours may be a more appropriate representation than a single deterministic compact model. The absence of universal structure would therefore still be scientifically informative because it could indicate that fabrication variation dominates the response, that several structural regimes coexist, or that hierarchical and distributional descriptions are more appropriate.
A sufficiently flexible learner can almost always produce some internal organization. The scientific test is whether that organization survives independent evidence.
Why this becomes a materials-science problem
This is where PSI modeling becomes particularly relevant to materials-driven electronics. The complete chain begins long before compact-model extraction. Fabrication conditions influence morphology, interfaces, defect populations, composition, layer thickness, conductive pathways, and other physical characteristics. Those features influence switching and transport behaviour, electrical characterization produces the resulting trajectories, and those trajectories are eventually used to construct compact models.
The objective is not to replace materials characterization. It is to determine whether those observations contain recurring structural organization that can be extracted, challenged, and compared with possible physical explanations. In printed memristive systems, for example, candidate structural signatures could include conductance-state organization, transition relations, characteristic relaxation scales, approximate symmetries, or recurrent dependencies. Such signatures can be assessed for consistency with candidate switching regimes without claiming that electrical measurements alone establish a unique microscopic mechanism.
From characterization to digital twins
The longer-term vision is an iterative experimental and computational workflow in which devices are fabricated and characterized, physics-informed models learn the measured dynamics under justified restrictions, and candidate structural organization is extracted and challenged using independent measurements, different operating conditions, chronology controls, repeated optimization runs, and eventually independent devices and fabrication batches. Only structures that remain supported should be converted into compact representations and translated into Verilog-A for circuit simulation.
The resulting digital twin would therefore be more than an accurate curve-fitting model. Its validity should be linked explicitly to the devices, operating regimes, and fabrication conditions under which its structure remains reproducible. Instead of asking only whether a model reproduces measured signals, we can also ask which part of its internal organization remains stable and over what range that organization can be trusted.
There is also a potential connection to sustainable electronics. PSI modeling does not make a material or device sustainable by itself, and lower prediction error is not an environmental metric. The possible contribution is more specific.
Any environmental benefit should ultimately be quantified rather than assumed.
Prediction is not the endpoint
Looking at the Ψ-family as a whole, I increasingly see the progression not as a sequence of individual neural architectures, but as a sequence of scientific questions. Ψ-NN asked whether physics-informed learning could reveal structured internal organization. Ψ-HDL asked whether discovered structure could become an executable compact representation. Ψ-xLSTM asked whether structure discovery could extend to temporal and high-frequency dynamics. Ψ-Vortex asked whether recurrent structure could be selected, challenged, compacted, and transferred into simulation with stronger validation controls.
The next question extends beyond any individual framework. Does the inferred structure survive the variability of real emerging electronic technologies? If it does, we gain evidence for compact representations that describe more than one fitted experiment. If it survives only within restricted operating conditions, those boundaries become part of the model. If it does not survive, variability itself becomes the scientific result that needs to be represented.
That is increasingly how I think about PSI modeling. It is not only a way to obtain another accurate neural surrogate, but a way to move from prediction toward the scientific testing of learned organization. Accuracy remains necessary, but it is no longer the final criterion. A discovered structure should also be challenged, assigned a validity range, and tested against the variability of the physical system from which it was learned. For emerging electronics, this could become especially important because materials, processing, device history, and fabrication are inseparable from the behaviour that we ultimately want to simulate.
Neueste Kommentare