Continual Learning of Dynamical Systems in Recurrent Neural Networks through Recyclable Unit Gating
Sima Hashemi, Daniel Durstewitz, Georgia Koppe
cs.LG
2026-09-30
CRUG gates and recycles RNN units so one network learns four dynamical systems sequentially with zero forgetting, reaching overall error 1.26 vs 3.37 for interleaved training.
Dynamical Systems Reconstruction (DSR) asks more of a model than next-step prediction: the model must evolve autonomously and reproduce the attractor geometry and temporal statistics of the observed system. For chaotic systems the two criteria come apart, since nearby trajectories diverge and accurate short-term forecasts say nothing about whether the dynamics were recovered.
Continual DSR (cDSR) adds a sequential constraint: systems arrive one at a time, and the model must keep earlier reconstructions intact while learning new ones. Recurrent networks are unusually fragile here, because parameters are applied repeatedly during free rollout and small updates can qualitatively change the generated dynamics over long horizons. Whether CL methods built for feedforward classification survive this stricter standard had no systematic answer. The paper benchmarks all three major families (parameter regularization, replay, parameter isolation) on the AL-RNN, a piecewise-linear, fully trainable and interpretable architecture, and introduces CRUG.
CRUG sits in the parameter-isolation family: each task gets its own subset of units, so interference is prevented by construction. Three mechanisms:
At inference no task-specific masks are needed: all learned systems run concurrently in one network, each read out through its own readout units. That separates CRUG from HAT-style methods that store per-task masks.
Main benchmark: Van der Pol → Lorenz-63 → Rössler → Chua learned in sequence (AL-RNN, 160 units, 80 nonlinear), 10 runs. Overall performance is the worst final task within each run; Dstsp is state-space divergence and DH the Hellinger distance between power spectra, lower is better. Non-isolation methods share all 160 units.
| Method | Overall Dstsp | Overall DH | Units committed |
| EWC | div. | div. | 160 |
| SI | 15.19 | div. | 160 |
| ER | 10.11 | 0.04 | 160 |
| GR | 7.39 | 0.05 | 160 |
| XdG | 17.70 | 0.14 | shared |
| CLNP | 3.81 | 0.14 | 123.5 |
| CRUG | 1.26 | 0.08 | 63 |
| Interleaved training | 3.37 | 0.05 | 160 |
CRUG finished all 10 runs; CLNP exhausted the free-unit pool in 3 of 10. CRUG's overall Dstsp is lower than interleaved joint training on all four systems at once (1.26 vs 3.37, p=0.038), mostly through better Rössler reconstruction. EWC and SI learn the tasks and then forget them; experience replay keeps the first two systems but degrades Rössler from 2.67 to 10.11. Alternative task orders and a second set of four chaotic systems (Blasius, Laser, Genesio–Tesi, Finance) reproduce the picture: there CRUG reaches Dstsp below 1.0 on all tasks with zero forgetting, committing about 60% of units and storing no replay data.
On real data, gait recordings from 12 subjects with Parkinson's, Huntington's, ALS and controls served as 12 sequential tasks for a 1024-unit AL-RNN: no measurable forgetting (median D̃stsp = 2.12) at roughly 48% capacity, and 27% better reconstruction than four group-specific models with 30% fewer committed units. On a suite of nine cognitive tasks, CRUG reached the highest final accuracy of all tested methods while using about 30% of capacity.
Transfer pays off when tasks share structure: on four Lorenz-63 variants differing only in ρ, every transfer variant beat the no-transfer condition on the reconstruction-capacity Pareto front, and the RMS ratio of transfer to within-task weights was significantly larger than on the heterogeneous sequence (p < 0.001).
For anyone modeling dynamics in neuroscience, ecology or climate, this is the first systematic CL comparison under the strictest standard, long-term autonomous fidelity in a fully trainable RNN, and the collective failure of EWC, SI and replay here is itself informative. CRUG's capacity economics are practical: more systems per network of the same size, no task routing at inference. The mechanism is not tied to AL-RNN; any recurrent architecture whose connectivity can be selectively constrained could adopt it. For the LLM crowd this is a small-model, mechanism-first paper; the transferable part is the design (unit-level allocation, recycling, one-way transfer), not a drop-in library.
The authors list four: task-incremental assumptions with known task boundaries and identities; recycling delays but does not prevent capacity exhaustion, since committed units are never released; the learned allocation should not be read as a system's intrinsic minimum capacity, as it depends on optimization, regularization and previously learned representations; and what exactly forward transfer moves between related systems remains unclear. Two more concerns from the results: the committed-unit pool is a hard ceiling, so scalability in task count is unaddressed, and all methods do far better on the cognitive tasks than on cDSR, which marks those results as a generalization check rather than the main claim.