The Converter Problem
A long-form essay on PINNs, ternary weights, ReRAM, and the seams between them
ReRAM crossbars compute dot products in the physics of the silicon. Getting the answer out costs more than the computation. This is the unsexy bottleneck deciding whether in-memory AI computing ships.
Three mathematical frameworks from the frugal-maths argument can stack, on paper, with no fundamental law preventing it. So who in the world is actually building it?
Here is the claim. A neural network, trained with physics constraints baked into its loss function—maths from the 1750s—with every single weight stored as a trit in {−1, 0, 1}—arithmetic from the 1950s—running on a resistive-memory crossbar where the matrix multiplication happens in the physics of the silicon rather than in a digital multiplier. Hardware from the 2010s.
Three frameworks. Three layers of the stack. Three entirely different moments in mathematical history.
They operate on different things. Physics-Informed Neural Networks (PINNs) constrain the training loss. Ternary quantisation constrains the weights. ReRAM constrains the deployment substrate. Because they touch different parts of the system, they don’t conflict. On paper, nothing prevents the combination.
The combination does not exist.
Not as a working system. Not as a published demo. Not as a line item on any research group’s public roadmap. What exists is three separate, highly mature research programmes, each producing real hardware or real models, none of which has yet been bolted to the others. The question is not whether the combination is mathematically coherent. It is. The question is whether anyone is actually close to building it, and if not, what precisely is stopping them.
The answer, it turns out, is more interesting than either a clean success story or a clean failure.
Three Constraints, Three Layers
To understand why the combination is viable in principle, you have to see exactly where each framework grips the system. The compatibility argument lives in the specifics.
A PINN embeds the governing equation of a physical system directly into the loss function. If you’re modelling fluid flow, the network is penalised not only for being wrong about the training data but for violating the Navier-Stokes equations. The governing equation is typically derived from the Euler-Lagrange equation of the system’s action functional—the variational calculus of the 1750s doing actual, structural work.
This operates entirely during training. It reshapes the loss landscape. It says absolutely nothing about how the trained model is stored, how many bits its weights occupy, or what silicon it runs on. It is agnostic to the deployment substrate.
Ternary quantisation constrains every weight in the network to {−1, 0, 1}. It operates on the representation of the model. Whether those weights were trained under a physics-informed loss or a standard cross-entropy loss is irrelevant to the quantisation process.
Yhe Straight-Through Estimator (STE) handles the backpropagation regardless of what the loss function looks like. Ternary is agnostic to the training objective.
ReRAM constrains the deployment hardware. A crossbar array computes matrix-vector multiplications by passing currents through variable resistors. It does not care whether those resistors represent 8-bit floats, 4-bit integers, or ternary trits. In fact, as established earlier in this series, a three-level ReRAM cell mapping to {−1, 0, 1} is arguably the most natural substrate for a crossbar, avoiding the wasted states of binary encoding or the complexity of multi-bit analogue encoding.
Three layers. Training objective. Weight representation. Physical hardware. They do not overlap. They do not interfere. There is no mathematical law that prevents a PINN-trained model from having ternary weights, and no physical law that prevents ternary weights from being mapped to a ReRAM crossbar.
On paper, the stack is clean.
Who Holds Which Piece
In practice, the stack is fragmented across different continents, different institutions, and different research communities that barely talk to each other.
The PINN people. George Karniadakis’s group at Brown University is the epicentre. Raissi, Perdikaris, and Karniadakis published the foundational PINN paper in 2019, and the Brown group has driven the field since. They work on the maths—adaptive weighting schemes to balance the physics loss and data loss, domain decomposition to handle complex geometries, extending the framework to stochastic PDEs and multi-physics problems. They train on GPUs. They export their models in PyTorch. They do not build hardware. They do not quantise to ternary.
The physics is their domain; the silicon is someone else’s problem.
The ternary people. The Chinese Academy of Sciences (CAS) has been a major hub for ternary network research since Li, Zhang, and Liu published Ternary Weight Networks in 2016. Microsoft Research’s BitNet team, also heavily based in Beijing, took the concept from convolutional networks into transformers with the BitNet b1.58 architecture in 2024.
These groups own the weight-representation layer. They care about the STE, about scaling factors, about keeping the first and last layers at full precision. They run on standard digital silicon—GPUs, FPGAs, custom ASICs. They do not use ReRAM, and with the exception of a few collaborations, they do not typically train physics-informed models. Their benchmark is ImageNet or WikiText-2, not the Navier-Stokes equations.
The ReRAM people. Huaqiang Wu’s group at Tsinghua University is among the most prolific ReRAM accelerator teams in the world. They fabricate chips. They design crossbar arrays, build the peripheral circuitry, fight the ADC problem, and publish energy-efficiency numbers that make digital silicon look lumbering. They need neural network weights to put on their crossbars, so they use whatever is available—usually standard INT8 or INT4 quantised models, occasionally binary weights.
They have demonstrated ternary-weight inference on ReRAM in small-scale proofs of concept. But they are hardware builders, not AI researchers. They do not design novel training regimes. They take an already-trained model, map it to the crossbar, and measure the hardware metrics.
Notice the gaps. The PINN community trains physics-aware models but doesn’t ternarise them. The ternary community ternarises models but doesn’t train them with physics constraints. The ReRAM community builds the hardware but relies on the other two for the software and the models.
The vertical integration required for the full combination does not exist within any single lab. It is distributed across the global research ecosystem like a jigsaw puzzle that nobody is trying to assemble.
The Closest to Existing
If you had to bet on where the combination emerges first, you would bet on China, and specifically on a collaboration between Tsinghua and CAS. The reasons are institutional and geographic, not just technical.
Tsinghua’s ReRAM fabrication infrastructure is among the most advanced in the world. They are not simulating crossbars; they are tape-outs, physical chips, measured results. CAS’s Institute of Computing Technology is a ten-minute subway ride away (or was, pre-pandemic) and has deep expertise in ternary networks. The BitNet team at Microsoft Research Asia is in Beijing. The talent pool overlaps. The informal networks overlap. If a PhD student at Tsinghua wants ternary weights for a ReRAM demo, they know who to call.
So what has actually been achieved?
Ternary-weight inference on ReRAM has been demonstrated. Not at LLM scale—think MNIST, CIFAR, small convolutional networks. But the principle is proven: you can program a ReRAM crossbar to three resistance states, map ternary weights, run an inference, and get an accurate result. The accuracy is slightly degraded by the analogue noise of the crossbar and the ADC’s quantisation, but it is in the right ballpark. This is the hardware-substrate layer combined with the weight-representation layer.
PINN inference on specialised hardware has also been demonstrated, though almost always on digital FPGAs or GPUs, not on analogue memory. The physics-informed loss doesn’t change the inference process—it only changes how the weights were trained. Once trained, a PINN is just a neural network, and it runs on whatever hardware you have.
What has not been demonstrated, as of mid-2025, is the full vertical stack: a physics-informed model, trained with a PDE constraint in the loss function, quantised to ternary weights during or after training, deployed on a ReRAM crossbar, solving a real-world physical simulation problem at an accuracy and efficiency that justifies the engineering effort.
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The missing piece is the integration, and the reason it’s missing is not mathematical. It is that no single funding stream, no single PI’s grant, no single company’s product roadmap requires it. Tsinghua’s ReRAM group gets funded to improve the hardware. Microsoft gets funded to improve LLM efficiency. Brown gets funded to advance scientific computing. The combination sits at the intersection of three fields, and intersectional research is always the last to get funded, because it doesn’t fit cleanly into any agency’s review panels.
The Obstacles That Aren’t Mathematical
Assume someone decides to build it. Assume a well-funded lab takes the three pieces and tries to integrate them. The obstacles they hit will not be in the maths. They will be in the engineering, and they will be ugly.
Fabrication yield on multi-level ReRAM. A binary ReRAM cell has two states: high resistance and low resistance. Distinguishing two states is straightforward; the gap between them can be made large. A ternary cell has three states: high, medium, low. The medium state must be reliably distinct from both neighbours. As devices age, as temperature fluctuates, as cycling causes material degradation, the medium state drifts. A trit that should read as 0 flips to −1 or 1.
In a language model, this is a gradual accuracy degradation. Annoying but survivable. In a PINN, it is potentially catastrophic. A PINN’s entire value proposition is that its outputs satisfy physical laws. A flipped trit in a critical weight could cause the model to predict a flow field that violates conservation of mass. The user—an engineer relying on the model for a structural simulation—might not notice the violation immediately. The model does not flag it; it simply outputs a physically impossible answer with confidence. The consequence of a hardware error in a physics-constrained model is qualitatively different from the consequence of the same error in a standard model, and the fabrication tolerances required to prevent it are correspondingly tighter.
The ADC, again. The converter problem doesn’t go away because the model is physics-informed. If anything, it gets slightly worse. PINN architectures often require higher numerical precision in their outputs than standard classification networks, because the output is a continuous physical field (a pressure distribution, a velocity vector, a temperature gradient) rather than a discrete class label. Coarse ADC quantisation introduces errors that look like numerical noise in the physical solution, and the network’s physics constraint—powerful as it is during training—does not explicitly protect against deployment-time hardware noise. Protecting against that would require training with hardware-noise injection, which is possible but adds another layer of training complexity.
The STE double whammy. This is the deepest problem, and it is the one that keeps the combination in the lab rather than in the field.
Training a PINN is already harder than training a standard neural network. The physics-residual loss and the data-fitting loss can have vastly different gradient magnitudes, creating a pathological optimisation landscape where the optimiser oscillates between satisfying the data and satisfying the physics. The community has developed elaborate techniques to handle this—adaptive loss weighting, gradient normalisation, curriculum learning—but it remains a finicky process.
Now add the Straight-Through Estimator. The STE is already an approximation: it lies to the optimiser about the gradient of the quantisation function. In a standard ternary network, this lie is manageable because the loss landscape, while complex, is relatively well-behaved. In a PINN, the loss landscape is already pathological. Lying to the optimiser on top of a pathological landscape is asking for trouble.
What happens in practice is that the training becomes dramatically more unstable. The physics loss and the data loss pull in different directions; the STE’s gradient approximation adds noise to an already noisy signal; the optimiser struggles to find a minimum that satisfies both the physics constraint and the ternary constraint simultaneously. Some researchers have reported that they can get it to converge by carefully tuning learning rates and loss weights, but the tuning is ad hoc, the success rate is low, and the resulting models have not been rigorously benchmarked against full-precision PINN baselines. The combination of STE and physics-residual loss is an open research problem, not a solved one.
The Use Case That Justifies the Whole Stack
Why bother? If the combination is this hard, if the obstacles are this ugly, if no single lab is incentivised to build it—why does the idea persist?
Because there is a use case where the combination is not just marginally better than the alternatives, but qualitatively necessary. And it lives at the edge of the physical world.
Consider a wind turbine blade. A modern offshore turbine blade is a hundred metres long, a composite structure subject to complex aerodynamic loads, gravitational cycling, temperature gradients, and salt-spray corrosion. Structural health monitoring—detecting micro-cracks and delamination before they become catastrophic failures—is a major engineering challenge.
The ideal monitoring system would run a physics simulation of the blade’s structural dynamics in real time, comparing the predicted behaviour (from a finite-element model driven by sensor data) against the actual behaviour (from strain gauges and accelerometers). Discrepancies indicate damage. The sooner you detect the discrepancy, the sooner you stop the turbine, and the cheaper the repair.
This simulation cannot run in a data centre. The latency of sending sensor data to shore, running a finite-element model, and sending the result back is too high for real-time monitoring of a rotating blade. It cannot run on a standard GPU on the turbine platform because the power budget is tight—everything on an offshore turbine runs on what the turbine itself generates, and power diverted to monitoring is power not sold to the grid. It cannot run a full-precision neural network because the memory bandwidth for a high-fidelity model exceeds what low-power digital silicon can provide.
A PINN-trained ternary model on a ReRAM chip addresses every constraint simultaneously. The PINN encodes the structural dynamics of the blade, so it doesn’t need a massive dataset to generalise—it knows the physics. The ternary weights compress the model, fitting it into a small memory footprint and reducing the memory-bandwidth bottleneck. The ReRAM crossbar eliminates the von Neumann tax, computing the simulation directly in the physics of the chip, at microwatt power levels.
The whole miserable engineering effort—the fabrication yield, the ADC overhead, the STE double whammy—is justified by a sensor package the size of a matchbox, bolted to a blade in the North Sea, running a structural simulation that keeps the turbine from tearing itself apart.
This is not a hypothetical. The individual components—the PINN for structural dynamics, the edge deployment requirement, the severe power constraint—exist in active research programmes within the wind energy and aerospace communities. What doesn’t exist is the integrated hardware. Yet..
The Road That Was Always There
The combination of PINN, ternary, and ReRAM is a thesis statement disguised as an engineering prospectus. It argues that the next generation of edge AI will not be achieved by making digital silicon slightly more efficient, but by restructuring the relationship between the maths, the representation, and the hardware.
The maths was never the problem. Lagrange’s formalism works. The ternary representation works. The physics of the crossbar works. The problem is the seams—the places where the maths meets the silicon, where the training loss meets the quantisation staircase, where the analogue current meets the digital converter. Engineering the seams is unglamorous work. It does not produce elegant papers. It produces yield reports, noise histograms, and failed training runs.
Brusentsov’s argument in 1958 was that ternary was mathematically superior and that the hardware could be built to match it. He was right on both counts. What he didn’t account for—and what almost nobody accounts for, in any era—is the gravitational pull of an installed base. Binary didn’t outcompete ternary on the merits. It outcompeted it on momentum.
The combination sits at a similar inflection point. The pieces exist. The labs that hold them are within shouting distance. The use case that demands the integration is real and growing. The question is whether the gravitational pull of the digital-GPU ecosystem—which is now unimaginably larger and more entrenched than the binary ecosystem of 1965 ever was—will prevent the pieces from being assembled before the window of relevance closes.
The road was always there. Brusentsov walked the first stretch of it. The question for the current generation of researchers is whether anyone will walk the last mile before the industry commits to another path.
Sources
- PINN foundational work: Raissi, Perdikaris, Karniadakis, “Physics-Informed Neural Networks,” Journal of Computational Physics 378, 2019. https://arxiv.org/abs/1711.10561
- Ternary Weight Networks: Li, Zhang, Liu, NIPS Workshop on EMDNN, 2016. https://arxiv.org/abs/1605.04711
- BitNet b1.58: Wang et al., arXiv 2402.17764, February 2024. https://arxiv.org/abs/2402.17764
- Tsinghua ReRAM accelerator work: multiple ISSCC and VLSI Symposium publications, Wu Huaqiang group, 2019—2025.
- Ternary inference on ReRAM: various prototype demonstrations in IEEE TCAS-I and JSSC, 2021—2024.
- PINN training instability: multiple survey papers, 2022—2024; Wang et al. on adaptive weighting.
- Cross-reference: Part 3, The Converter Problem, on the ADC bottleneck that constrains any ReRAM deployment.
- Cross-reference: Part 4, What Multiplication Actually Costs, on the energy economics that make the combination worth attempting.
