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Optimal Control Operator Perspective and a Neural Adaptive Spectral Method
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Proposes a one-pass instance-to-control-function operator learner (NCO) implemented by the Neural Adaptive Spectral Method (NASM), a neural operator with adaptive basis and time-dependent coefficients, with approximation error bounds and experiments on synthetic control systems and a real planar pushing dataset.
Problem
Solving optimal control problems efficiently and reusably without known dynamics, iterative optimization at test time, or discretization-limited control functions; existing two-phase neural solvers suffer from inconsistency and distribution shift.
Approach
Treat OCP solving as learning an instance-to-control operator and implement it with NASM, a generalized spectral method whose coefficients depend on both instance and time and whose basis functions are adaptively parameterized by the instance and time, enabling single-pass, continuous-time control outputs.
Key results
- On Quadrotor, NASM achieves 6.50e-5 s/instance vs 9.23e-2 s for DM (over 1000x; text claims over 6000x speedup) with ID MAPE 6.17e-6 and OOD MAPE 1.21e-4, the best among neural models.
- On the real Pushing dataset, NASM achieves the best ID MAPE (0.108) and second-best OOD MAPE (0.137), slightly behind DON (0.134).
- Under extreme OOD shift with fine-tuning, NASM achieves OOD MAPE of 1.02e-5 to 3.07e-5 on Quadrotor, best or near-best among neural models.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use NASM instead of DM | Quadrotor control · Quadrotor | Time (sec./instance): 9.23 \times 10^{-2} -> 6.50 \times 10^{-5} | Helped |
| use NASM instead of the strongest neural baseline, DON | Quadrotor control · Quadrotor | ID MAPE: 4.09 \times 10^{-5} -> 6.17 \times 10^{-6} | Helped |
| use NASM instead of PDP | Quadrotor control · Quadrotor | OOD MAPE: 1.16 \times 10^{-4} -> 1.21 \times 10^{-4} | Mixed |
| use NASM instead of DON | Pushing control · Pushing | ID MAPE: 0.117 (\pm 0.006) -> 0.108 (\pm 0.006) | Helped |
| use NASM instead of DON | Pushing control · Pushing | OOD MAPE: 0.134 (\pm 0.008) -> 0.137 (\pm 0.007) | Hurt |
| add adaptive basis parameters (ablation) | NASM · Quadrotor | ID MAPE: 7.86 \times 10^{-6} -> 6.17 \times 10^{-6} | Helped |
| add non-static, time-dependent coefficients (ablation) | NASM · Quadrotor | OOD MAPE: 5.83 \times 10^{-4} -> 1.21 \times 10^{-4} | Helped |
| replace adaptive basis and non-static coefficients with a fixed Fourier basis and static coefficients (ablation) | NASM · Quadrotor | ID MAPE: 6.17 \times 10^{-6} -> 9.59 \times 10^{-2} | Hurt |
| replace adaptive Fourier basis with adaptive Chebyshev basis (ablation) | NASM · Quadrotor | ID MAPE: 6.17 \times 10^{-6} -> 6.36 \times 10^{-6} | No clear effect |
| replace summation with neural aggregation (ablation) | NASM · Quadrotor | ID MAPE: 6.17 \times 10^{-6} -> 7.13 \times 10^{-6} | Hurt |
| replace MLP backbone with a convolutional backbone (ablation) | NASM · Quadrotor | OOD MAPE: 1.21 \times 10^{-4} -> 4.95 \times 10^{-3} | Hurt |
Surprising findings
- Fixing the basis to Fourier in the SNO variant dramatically increases error on non-periodic Quadrotor control (MAPE 9.59e-2), whereas NASM is insensitive to basis choice.
Limitations
- Does not specify forms of problem instances (ODE/PDE-constrained) or model specific problem structure
- Two-phase and neural baselines are sensitive to distribution shift; NASM's OOD error degrades substantially under extreme shifts without fine-tuning
- Encoder error assumed zero in theoretical analysis
Experimental setup
Datasets: Synthetic control systems (Quadrotor and others) generated with Direct Method solutions Pushing dataset (ABB IRB 120 real-world robot pushing)
Benchmarks: Quadrotor (ID/OOD MAPE) Synthetic control systems ID/OOD benchmarks Pushing dataset ID/OOD MAPE
Metrics: MAPE inference time per instance
Baselines: Direct Method (IPOPT) PDP DeepONet MLP FNO GEN SNO
Builds on: DeepONet Fourier Neural Operator Spectral Neural Operator Pontryagin Differentiable Programming
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