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ZNO: Stable Rational Neural Operators in the Z-Domain for Discrete-Time Dynamics
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Introduces ZNO, a causal neural operator built from stable low-rank MIMO rational filters parameterized directly by poles in the z-plane, evaluated on synthetic and public discrete-time system-identification tasks against FNO, S4D and classical NARMAX.
Problem
Operator learning methods are mostly tailored to continuous-time problems, while many system-identification tasks are intrinsically discrete-time; stability of learned discrete-time rational dynamics needs an appropriate parameterization.
Approach
Parameterize each latent rank channel as a causal rational filter with complex conjugate poles constrained to the unit disk by a smooth sigmoid map and residues, combined with low-rank channel mixing, an optional FIR branch, and a fused scan kernel.
Key results
- On the resonant ARMA task under matched budget ZNO reaches rel L2 0.203 vs 0.386 for S4D and 0.515 for FNO (5 seeds).
- ZNO has the lowest mean error in all five difficulty bins, e.g. 0.065 vs 0.215 (S4D) on the shortest-memory bin.
- On public benchmarks ZNO has the lowest mean error on Silverbox (0.021), Wiener-Hammerstein (0.003) and CED (0.150), but S4D wins cascaded tanks and a classical NARMAX model wins EMPS (0.080).
Surprising findings
- Isomorphic s-plane parameterization of the same low-rank rational layer performs indistinguishably from the z-plane version, so the coordinate choice alone does not drive gains.
- ZNO shows a large validation/test gap on EMPS (0.227 val vs 1.041 test), and a single divergent seed inflates the IIR matched-budget result.
Limitations
- Under matched budget ZNO is not uniformly best; S4D wins on the IIR cascade at matched budget and cascaded tanks.
- Large validation/test gap and poor performance on EMPS relative to classical NARMAX.
- Wall-clock comparisons are confounded by a fused Triton kernel for ZNO versus plain PyTorch baselines.
Experimental setup
Datasets: Silverbox Wiener-Hammerstein Cascaded tanks CED EMPS synthetic resonant ARMA sixth-order IIR cascade nonlinear NARX
Benchmarks: resonant ARMA sixth-order IIR cascade nonlinear NARX Silverbox Wiener-Hammerstein Cascaded tanks CED EMPS
Metrics: relative L2 error
Baselines: FNO S4D LNO SysIdentPy AOLS/FROLS NARMAX
Builds on: Fourier Neural Operator Laplace Neural Operator S4D
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