A Physics-Regularized Neural Surrogate Framework for Printed Memristors

My research paper „A Physics-Regularized Neural Surrogate Framework for Printed Memristors“ was published today in the IEEE Access journal (IF: 3.6).

I had the idea for this paper initially while working on the „Print the Brain“ project at TU Chemnitz in 2024, and thought about it in my free time while at ERCEA in Brussels until end of April 2025. In the summer of 2025, I finally had more time to make progress on it. My approach in this work combines two areas I find fascinating: the complex physical reality of printed electronics and the elegant framework of physics-informed machine learning. It’s a new way to model printed memristors, those flexible memory devices that could one day power brain-like computing hardware, and I wanted to share what I’ve built and learned.

Printed memristors are exciting because they promise inexpensive, flexible neuromorphic chips that could potentially be rolled out like newspapers. But they come with a catch: they’re wildly variable.

Traditional compact models, the kind circuit designers use in SPICE simulations, struggle with this. They often assume idealized physics and relatively uniform parameters, which works fine for pristine silicon devices but falls apart for printed ones.

That’s where physics-informed machine learning comes in. While classical Physics-Informed Neural Networks (PINNs) are often described as enforcing governing equations directly (I discovered and used PINNs for the first time a few years ago while working as a scientific researcher in the KI-Wissen project at the German Aerospace Center), my approach is best described as a PINN-inspired, physics-regularized neural surrogate (PRNS). More exactly, I train a neural model on data, and I add physics-based regularization terms that strongly bias it toward physically plausible behavior instead of letting it learn a purely black-box mapping.

Here’s how I structured it. The network takes voltage and an internal state variable as inputs, and predicts current and how that state evolves. The twist is in the loss function. I break it down into two parts: one penalizes mismatches with measured (or generated) data (the usual supervised learning part), and the other penalizes deviations from physical behavior. In the low-voltage regime, I regularize toward Ohmic conduction (approximately linear I–V). In the intermediate regime, I regularize toward space-charge-limited current (SCLC) (square-law dependence). I also include a state-dynamics consistency term to capture the memory effect and keep the internal evolution coherent. Interface effects such as Schottky emission are described in the model but not enforced as hard constraints, reflecting the high variability of printed metal-polymer interfaces.

What makes this suited explicitly for printed devices is how I handle variability and noise. During training, I don’t just feed the network clean curves. I perturb physical parameters within a small bound (to emulate device-to-device differences such as thickness or mobility) and I also inject Gaussian noise into voltage/current signals (to emulate measurement conditions). This forces the model to learn a family of plausible device behaviors rather than memorizing one pristine curve, and it pushes the predictions to remain stable under realistic corruption.

I validated this first on synthetic data, which might sound like cheating, but it’s standard practice in compact modeling: you create a controlled testbed where you know the “ground-truth” generative physics, and then verify that your model can recover it. My synthetic dataset was calibrated to literature on flexographically printed Ag/PMMA:PVA/ITO devices, including realistic variability and cycle-to-cycle noise. Across 20,000 voltage–current points from multiple virtual devices, my PRNS achieved a relative RMSE (RRMSE) of 0.0633, compared to 0.2730 for the widely used VTEAM phenomenological model. That’s a 4.31× improvement in this benchmark.

But numbers only tell part of the story. The real test is whether training is stable and repeatable. To assess stability and reproducibility, I repeated the full training and evaluation procedure across eight different random seeds. The optimization landscape is clearly non-convex: performance varied substantially depending on initialization. The best run achieved an RRMSE of 0.033, while the worst reached 0.202, indicating multiple local minima of differing quality. Importantly, even the worst-performing instance still outperformed the VTEAM baseline (0.273). Across all eight seeds, the median RRMSE was 0.065 with an interquartile range of 0.046–0.106, which is why I report median and IQR rather than best-case results. This highlights both the strength of the approach and the remaining challenge of initialization sensitivity. Rather than a fundamental limitation, this behavior points to an opportunity: combining physics-regularization with better initialization, ensembles, or curriculum learning could significantly improve convergence consistency while preserving the model’s physical interpretability.

I also tested noise robustness, ramping Gaussian noise on the test set from 1% to 10%. Even at 10% noise, which is quite severe, the error only degraded by about 3%.

It’s important to be precise about what the physics terms contribute. In my experiments, a purely data-driven network of the same architecture can sometimes achieve a lower raw fit error on held-out synthetic data. The value of physics-regularization is that it dramatically improves physical consistency (large reductions in physics residuals) and strengthens robustness and extrapolation behavior, which are exactly the failure modes that tend to show up when you move from clean benchmarks to messy printed-electronics reality.

To test whether this generalizes beyond my synthetic case, I digitized I–V curves from three different printed memristor technologies: inkjet-printed IGZO, aerosol-jet MoS₂, and paper-based MoS₂/graphene devices. I benchmarked against four established compact models: VTEAM, Yakopcic, Stanford-PKU, and MMS. My surrogate achieved the lowest error on the MoS₂ and paper devices. IGZO was the difficult case in this configuration: oxide-oriented baselines (notably VTEAM and Stanford-PKU) fit those digitized curves better. The broader takeaway is that the framework can generalize across very different technologies, but some material systems may benefit from tailoring the regime assumptions or the regularization design.

The circuit integration part was particularly satisfying to develop. I exported the trained surrogate as a lookup table (LUT): a 500×50 grid over voltage and internal state, i.e., 25,000 points that a SPICE-like simulator can interpolate. I then simulated a standard one-transistor–one-resistor (1T1R) memory cell, comparing the surrogate’s dynamics with those of VTEAM under identical write pulses. VTEAM, being threshold-based, switches abruptly from off to on in about 18.6 ns. My surrogate exhibits more gradual switching over about 27.5 ns, which is valuable for analog neuromorphic applications where you want to tune conductance in smaller increments rather than flip a hard digital bit. The surrogate also predicted 29% lower write energy (94.2 pJ versus 133.6 pJ) in that setup, consistent with capturing partial switching behavior rather than forcing an abrupt transition.

Now, regarding limitations: everything I’ve described is based on simulations. I didn’t fabricate new devices or measure fresh I–V curves. My primary validation uses synthetic data calibrated to the literature, and my external validation uses digitized curves rather than raw measurements. This was a deliberate choice: it let us test the framework comprehensively and reproducibly without lab access, but it means the real proof still requires experimental collaborators running the framework on raw device data. There’s also initialization sensitivity. Achieving the best result (0.033 RRMSE) required a favorable random seed, while the representative configuration (0.0633) reflects a more typical outcome. Ensemble methods, better initialization, or curriculum-style training could reduce this variability. And while I demonstrated a practical path to circuit use via LUTs, a full Verilog-A implementation and rigorous solver convergence testing in commercial EDA tools remains future work.

Still, I’m excited about the potential. This framework bridges the gap between detailed physical modeling (which can be slow and parameter-heavy and purely data-driven approaches) which can fit curves but often extrapolate poorly and violate physical intuition. By embedding what we know about conduction mechanisms while allowing the network to learn what we don’t, we can obtain models that are both accurate in practice and robust to the variability inherent in printed electronics.

Neuromorphic computing needs to transition beyond pristine lab devices to practical, manufacturable hardware. Printed memristors could be part of that future, but only if we can model them well enough to design circuits with confidence. This PRNS framework is a step toward that goal, not the final answer, and provides a solid foundation that I’m sharing with the hope that others will build on it, test it, and help make printed neuromorphic electronics a reality.