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The success of large-scale AI relies on the phenomenon of double descent, where overfitting first increases and then counterintuitively decreases as the network grows relative to the amount of training data and becomes overparameterized. In this overparameterized regime it is easy to find good solutions that do not overfit. However, double descent has not previously been observed in physical neural networks, which are attracting increasing attention but remain small-scale. Here we demonstrate double descent in a decentralized analog network of self-adjusting resistive elements. This system trains itself and performs tasks without a digital processor, offering potential gains in energy efficiency and speed—but must inevitably confront component nonidealities. We find that standard training protocols fail to yield double descent, but a modified protocol that accommodates generic imperfections succeeds, qualitatively matching digital neural network simulations. Our findings show that analog physical neural networks, if appropriately trained, can reproduce important facets of overparameterized digital learning. This result is critical if such networks are to be scaled up to useful sizes. 

Abstract: In principle, computation with analog devices is faster, more compact, and lower power than using digital devices. However, compelling analog systems to perform precise computations, especially with many parameters, has long been a challenging goal, and stands in the way of low-power low-latency AI. Analog electronic contrastive local learning networks (CLLNs) learn to perform computations with many parameters, and in this dataset we include experimental demonstrations and companion simulations of such a system exhibiting the phenomenon of double descent, wherein raising the ratio of parameters to training datapoints first decreases then increases the generalization of a learned function. This phenomenon is a byproduct only of precise, many parameter optimization, and an indication that analog systems can exhibit essential properties of deep learning systems. Included here are data and scripts required to generate figures 2-4 and S1-3 of the manuscript titled "Double descent without digital computation". TechnicalInfo: # Data and code from: Double descent without digital computation [https://doi.org/10.5061/dryad.8cz8w9h6p](https://doi.org/10.5061/dryad.8cz8w9h6p) Data from the double-descent experiments on a nonlinear analog learning network, with additional simulations on a digital network. ## Description of the data and file structure Unzip PNASscripts2026.zip and run all scripts from this folder (`PNASscripts2026/`). Helpers are in `src/`; all input data are in `data/`. | Figure | Description | Script | Flags | | ------- | ------------------------------------------------------------------------------- | ----------------- | --------------------------------------------------------- | | Fig. 2 | Train and test error vs γ on the analog network | `MakeFigs2and3.m` | `DATASET = 'doubledescentZ_22'` | | Fig. 3 | Negative-class output histograms, hinge loss, and example decision maps | `MakeFigs2and3.m` | `DATASET = 'doubledescentZ_22'`; `NEGATIVE_CLASS = true` | | Fig. 4 | Simulated train/test loss (2 features) compared with experiment | `MakeFig4.ipynb` | | | Fig. S1 | Same as Fig. 2 but without overclamping technique (no double descent) | `MakeFigs2and3.m` | `DATASET = 'doubledescentZ_43'` | | Fig. S2 | Same as Fig. 3 for the positive class | `MakeFigs2and3.m` | `DATASET = 'doubledescentZ_22'`; `NEGATIVE_CLASS = false` | | Fig. S3 | Same as Fig. 4 but for more (16) input features (peak moves to predicted value) | `MakeFigS3.ipynb` | | In the python notebooks, `USE_RAW = False` (default) loads pre-averaged simulation means; `USE_RAW = True` averages from the per-run `.csv.gz` files. `RUN_SWEEP = True` will resimulate (slow). src/ {MATLAB and Python helpers: Experiment2.m, network_project_superclass2.m, nonlinearClassificationMap6.m, makeErrorBar.m, makePlotPrettyNow.m, makeColormap.m, train_data.py, numerics.py} data/{CSV, MAT, and gzipped simulation files listed below} ### Experimental CSVs (`doubledescentZ_22*.csv`; `doubledescentZ_43*` for Fig. S1) One row per experimental run unless noted. Training arrays are padded with NaN where a run has fewer than the maximum number of training points. * `doubledescentZ_22Dates.csv`: run ID (date-time string). * `doubledescentZ_22TrainLabel.csv`, `TrainIn1.csv`, `TrainIn2.csv`, `TrainOutput.csv`: training labels, two analog inputs, and measured outputs. * `doubledescentZ_22TestLabel.csv`, `TestIn1.csv`, `TestIn2.csv`: shared test labels and inputs (one row). * `doubledescentZ_22TestOutput.csv`: measured test outputs (rows = runs, columns = test points). Labels are in 0–1 measurement units with magnitude ρ = 0.005; multiply by NODEMULT ≈ 0.45 to convert to volts. ### Other experimental files `doubledescentZ_22_data.mat`: task coefficients and related constants (`coeff`, `TEST`, …) used to draw the decision line in Fig. 3. `experiment_doubledescentZ_22_TRA=*.mat`: nine example runs (training size TRA = 2, 6, or 32) stored as an `experiment` object (class in Experiment2.m). Fields include: * TRA: training set size * TST: test set size * MES: number of measurement steps * SOR, TAR: number of source (input) and target (output) nodes * NODEMULT, GATEMULT: convert 0–1 stored values to volts * TRAIN, TEST: training/test sets (sources then targets are rows) * HorizontalCapacitors, VerticalCapacitors: gate voltages vs. measurement step (0–1; × GATEMULT for volts) * TestFreeState: free-state node voltages vs. measurement step and test point (0–1; × NODEMULT for volts) ### Simulation CSVs Columns: `train_size` (M), `params` (P), `features`, `hinge_loss_train`, `hinge_loss_test`, `lambda`, `function` (`tanh` or `relu`). Raw files also have `replicate`. `SimulatedData_both_07pct_02F_withconst5.28.csv`: Fig. 4 means (2 input features). Matching raw runs are split across `*_raw_tanh_P1-5.csv.gz`, `*_raw_tanh_P6-10.csv.gz`, `*_raw_relu_P1-5.csv.gz`, `*_raw_relu_P6-10.csv.gz`. `SimulatedData_both_07pct_16F_withconst5.4.csv`: Fig. S3 means (16 input features). Matching raw runs are `*_raw_tanh.csv.gz` and `*_raw_relu.csv.gz`. ## Sharing/Access information Data were taken in experiments and simulations described in the manuscript. ## Code/Software MATLAB figures: MATLAB R2020b or later. Python figures: Python 3 with numpy, pandas, matplotlib, scipy, and tqdm. 

As the size and ubiquity of artificial intelligence and computational machine learning models grow, the energy required to train and use them is rapidly becoming economically and environmentally unsustainable. Recent laboratory prototypes of self-learning electronic circuits, such as “physical learning machines,” open the door to analog hardware that directly employs physics to learn desired functions from examples at a low energy cost. In this work, we show that this hardware platform allows for an even further reduction in energy consumption by using good initial conditions and a new learning algorithm. Using analytical calculations, simulations, and experiments, we show that a trade-off emerges when learning dynamics attempt to minimize both the error and the power consumption of the solution—greater power reductions can be achieved at the cost of decreasing solution accuracy. Finally, we demonstrate a practical procedure to weigh the relative importance of error and power minimization, improving the power efficiency given a specific tolerance to error. 

Abstract Seismic moment and rupture length can be combined to infer stress drop, a key parameter for assessing earthquakes. In natural earthquakes, stress drops are largely depth‐independent, which is surprising given the expected dependence of frictional stress on normal stresses and hence overburden. We have developed a transparent experimental fault that allows direct observation of thousands of slip events, with ruptures that are fully contained within the fault. Surprisingly, the observed stress drops are largely independent of both the magnitude of normal stress and its heterogeneity, capturing the independence seen in nature. However, we observe larger, normal stress‐dependent stress drops when the fault area is reduced, which allows slip events to frequently reach the edge of the interface. We conclude that confined ruptures have normal stress independent stress drops, and thus the depth‐independent stress drops of tectonic earthquakes may be a consequence of their confined nature. 

Classically, the quantity of contact area AR between two bodies is considered a proxy for the force of friction. However, bond density across the interface—quality of contact—is also relevant, and contemporary debate often centers around the relative importance of these two factors. In this work, we demonstrate that a third factor, often overlooked, plays a significant role in static frictional strength: The spatial distribution of contact. We perform static friction measurements, μ, on three pairs of solid blocks while imaging the contact plane. By using linear regression on hundreds of image-μ pairs, we are able to predict future friction measurements with three to seven times better accuracy than existing benchmarks, including total quantity of contact area. Our model has no access to quality of contact, and we therefore conclude that a large portion of the interfacial state is encoded in the spatial distribution of contact, rather than its quality or quantity 

Standard deep learning algorithms require differentiating large nonlinear networks, a process that is slow and power-hungry. Electroniccontrastive local learning networks(CLLNs) offer potentially fast, efficient, and fault-tolerant hardware for analog machine learning, but existing implementations are linear, severely limiting their capabilities. These systems differ significantly from artificial neural networks as well as the brain, so the feasibility and utility of incorporating nonlinear elements have not been explored. Here, we introduce a nonlinear CLLN—an analog electronic network made of self-adjusting nonlinear resistive elements based on transistors. We demonstrate that the system learns tasks unachievable in linear systems, including XOR (exclusive or) and nonlinear regression, without a computer. We find our decentralized system reduces modes of training error in order (mean, slope, curvature), similar tospectral biasin artificial neural networks. The circuitry is robust to damage, retrainable in seconds, and performs learned tasks in microseconds while dissipating only picojoules of energy across each transistor. This suggests enormous potential for fast, low-power computing in edge systems like sensors, robotic controllers, and medical devices, as well as manufacturability at scale for performing and studying emergent learning.