Sequential Learning and Catastrophic Forgetting in Differentiable Resistor Networks

TL;DR

This paper studies how differentiable resistor networks learn sequentially, revealing the physical and topological factors influencing catastrophic forgetting and proposing insights into continual learning in physical systems.

Contribution

It demonstrates how task conflict and network topology affect forgetting in differentiable resistor networks, linking physical reconfiguration to learning challenges.

Findings

Forgetting correlates with local conductance changes on high-current edges.

Uniform anchoring reduces forgetting but increases final loss on new tasks.

Topology influences the balance between forgetting and adaptation.

Abstract

Differentiable physical networks provide a simple setting in which learning can be studied through the interaction between trainable parameters and physical equilibrium constraints. We investigate sequential learning in differentiable resistor networks governed by Kirchhoff's laws. Although individual input--output mappings can be learned by gradient-based adjustment of edge conductances, sequential training on conflicting tasks produces catastrophic forgetting. We show that forgetting is controlled by task conflict and by the degree of adaptation to the new task. Uniform anchoring and normalised gradient-weighted anchoring reduce forgetting only by increasing the final loss on the new task, giving a clear forgetting--adaptation trade-off. We also show that forgetting is associated with localised conductance changes on high-current edges, giving a physical interpretation as reconfiguration of dominant transport pathways. Broader random-task ensembles show that the strongest forgetting occurs when the second task reverses the output ordering imposed by the first task. Finally, comparisons across Erd\H{o}s--R\'enyi, small-world, scale-free, and random-geometric graph ensembles show that topology changes the forgetting--adaptation balance. These results position differentiable resistor networks as compact, physically interpretable testbeds for studying continual learning in tunable matter.