Untrainable elements determine what physical learning remembers

Physical learning rules such as equilibrium propagation (EP), coupled learning (CL), and adjoint coupled learning (AL) train resistive networks through local measurements. The learned function is decided by where on the solution manifold training lands. Two properties could decide it, and they have not been separated: the circuit's invariance under rescaling every conductance, and the rule's conservation of the mass K = (1/2) sum_e kappa_e^2. We separate them. When every element is trainable, all three vector fields are homogeneous in the conductances, so the initialization scale is provably inert. An element the rule does not adjust breaks that homogeneity whatever its constitutive law. Across twenty topologies the learned function moves with the initialization scale by a median of twelve percent with fixed rectifiers and eight with fixed linear resistors, against 3e-8 when every element is trainable; a single fixed rectifier produces the whole effect. The conservation law is not what protects the function: AL, which we prove dissipates the mass at exactly twice its own loss, remembers its initialization as strongly as the rules that conserve it, and the memory survives in runs where K is conserved to 1e-4. Raising the fixed-element count from one to eight multiplies the conservation drift by five thousand and leaves the memory unchanged, while the all-trainable circuit under AL drifts comparably and remembers nothing. What the rule's conservation structure does control is solution quality: at matched training loss AL is worse than EP and CL in four of six small circuits, by a median of three to seven percent, though the ordering is not stable across checkpoints and does not reproduce at fifty nodes. Physical learning therefore carries two independent inductive biases, one belonging to the circuit and one to the rule, and only the first is a memory of how the device was built.

It's Much Easier for Neural Networks to learn Game of Life Dynamics with the Right Activation Function: Polynomial Kolmogorov-Arnold Networks

Previous work has found a gap between the scale of neural networks that reliably learn Conway's Game of Life, and minimal networks capable of representing the classic cellular automaton with hard-coded parameter values. Viewing neural network learning as a search process suggests a dependence on networks large enough to contain sub-networks with lucky initializations (sometimes known as 'winning tickets') that actually learn the task. In this work, we reorient our perspective from discovering Life rules as a search problem back to a learning problem, and reason that with fitting inductive biases, the problem should be much more amenable to minimal networks. We find that network variants with several alternative activation functions meaningfully outperform the default choice of Rectified Linear Units, and in particular, that a 2nd degree polynomial activation function consistently learns Life dynamics with or without the benefit of learning neural weights. Our results provide an informative demonstration of the benefits of matching learning to the task at hand and challenge the easy default choice of scale for all problems. In particular, we advocate for the use of cellular automata as simple test domains for developing strategies that can benefit machine learning for science, physics-based deep learning, and interpretable machine learning.

Harnessing cortical geometry, wiring, and function as inductive biases for recurrent neural networks

How the wiring and functional organization of cortex shape recurrent computation remains a central question in both neuroscience and machine learning. Here, we leverage data released through the Machine Intelligence from Cortical Networks (MICrONS) program--a functional connectomics resource spanning multiple areas of mouse visual cortex, in which dense calcium imaging is co-registered with high-resolution electron microscopy reconstruction from the same animal--to build biologically grounded recurrent neural networks. Using neuronal spatial coordinates, anatomical connectivity, and function-derived relationships from nearly 12,000 coregistered excitatory neurons, we initialize recurrent weights and impose communication-aware spatial constraints during learning. Across three cognitive decision-making tasks, networks constrained by cortical structure and function consistently outperform baseline and partially constrained models. Functional weight initialization provides the largest gain, while real spatial embedding yields robust additional improvements across conditions. These biologically grounded networks also develop low-entropy, modular, and small-world organization, and retain strong performance even when recurrence is restricted to positive weights. Together, our results show that the machinery of cortex--its geometry, wiring, and functional structure--can be harnessed as a powerful inductive basis for building recurrent networks that learn more effectively while converging toward key organizational principles of biological computation.