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Nyström-Accelerated Primal LS-SVMs: Breaking the $O(an^3)$ Complexity Bottleneck for Scalable ODEs Learning
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Proposes Nyström-accelerated primal LS-SVMs (NLS-SVMs) that turn ODE solving into a primal problem over an explicit m-dimensional Nyström feature map and its derivatives, cutting the O(an^3) cost of classical LS-SVM ODE solvers.
Problem
Classical kernel-based LS-SVM ODE solvers require solving dense systems whose cost scales as O(a n^3) in the number of temporal discretization points n (a=1 linear, 27 nonlinear), which is prohibitive for long-time or fine-grained ODE simulations.
Approach
Use Nyström landmarks to build an explicit finite-dimensional feature map φ(t) ∈ R^m and its analytical derivatives from RBF kernel derivatives, so the ODE LS-SVM is solved in primal space with m-dependent cost, with Newton iterations for nonlinear ODEs.
Key results
- NLS-SVMs runs 10 to 6426 times faster than LS-SVMs and PINNs depending on the problem and n, per the paper's speedup table.
- Accuracy vs LS-SVMs within 0.13% relative MAE, RMSE, and L-infinity difference across problems.
- Versus PINNs, RMSE improves by up to 72%; scales to n=5x10^4 time steps where LS-SVMs and PINNs ran out of memory, with runtimes of 17.5 to 460 s.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use NLS-SVMs instead of RK4 | ODE solver · Problem 3 | MAE: 3.87×10^{-7} -> 2.06×10^{-8} | Helped |
| use NLS-SVMs instead of RK4 | ODE solver · Problem 6 | MAE: 1.59×10^{-2} -> 8.89×10^{-4} | Helped |
| use NLS-SVMs instead of EAB | ODE solver · Problem 6 | MAE: 1.28×10^{-6} -> 8.89×10^{-4} | Hurt |
| use NLS-SVMs instead of EAB | ODE solver · Problem 11 | MAE: 1.19×10^{-2} -> 2.36×10^{-9} | Helped |
| use equidistant rather than random landmark sampling (ablation) | NLS-SVMs · Problem 1 | RMSE: 1.06×10\textsuperscript{-3} -> 9.47×10\textsuperscript{-4} | Helped |
| use equidistant rather than leverage score landmark sampling (ablation) | NLS-SVMs · Problem 1 | Time/s: 0.64 -> 0.55 | Helped |
| increase Vanilla PINN depth from three to eight layers (ablation) | Vanilla PINN · Problem 1 | Time: 672 s -> 1275 s | Hurt |
| increase Vanilla PINN depth from three to eight layers (ablation) | Vanilla PINN · Problem 1 | MAE: 4.45×10^{-3} -> 2.32×10^{-3} | Helped |
| replace Vanilla PINN with Fourier Feature PINN (ablation) | PINN · Problems 3, 11, and 15 versus Problems 4, 7, and 16 | error (MAE, RMSE, and L∞) | Mixed |
| use NLS-SVMs instead of PINN | ODE solver · Problem 4 | RMSE: ↓7.20 × 10^{1} | Helped |
| use NLS-SVMs instead of LS-SVMs | ODE solver · Problem 4 | Speedup: ↑ 370 | Helped |
| use NLS-SVMs instead of PINN | ODE solver · Problem 16 | Speedup: ↑ 6426 | Helped |
Surprising findings
- The paper states its nonlinear-ODE complexity remains O(n^3), so the claimed O(m) reduction does not cover nonlinear ODEs.
- Equidistant landmark sampling matched or beat random and leverage-score sampling in most cases.
Limitations
- Nonlinear ODE complexity remains O(n^3) per Newton iteration, which the authors state as a primary limitation.
- Method is validated only on ODE benchmarks, not PDEs or other kernel tasks.
- Hyperparameters (gamma, sigma) are tuned per problem class and convergence theory relies on assumed Lipschitz and bounded-inverse conditions.
Experimental setup
Datasets: sixteen benchmark ODE problems
Benchmarks: sixteen benchmark ODEs (first-, second-, fourth-order)
Metrics: MAE RMSE L-infinity error R^2 runtime
Baselines: LS-SVMs PINN (vanilla, three-layer) Fourier Feature PINN RK4 explicit Adams-Bashforth
Builds on: LS-SVMs Nyström method Mercer's theorem
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