Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Learning to Solve Differential Equation Constrained Optimization Problems
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Proposes DE-OP, a dual-network learning framework that pairs a proxy optimizer for decisions with a neural differential equation surrogate for state dynamics, trained via primal-dual learning, for near real-time DE-constrained optimization.
Problem
DE-constrained optimization (decisions that control ODE/SDE system dynamics) is computationally expensive for near real-time use; existing proxy optimizers ignore system dynamics and violate them.
Approach
Couple a proxy optimizer network (decisions) with neural-DE surrogates (state trajectories) so that the dynamic equations are treated as implicit equality constraints in a Lagrangian primal-dual training loss, enabling end-to-end differentiable training.
Key results
- On IEEE 57-bus, DE-OP reports zero stability violations while MSE, LD and DC3 have 23.1% to 28.6% violations.
- On portfolio optimization with n=50, DE-OP with neural-SDE achieves a 9.11% optimality gap versus 21.17% for LSTM and over 100% for static proxy optimizers, reported as up to 25x better decisions.
- Neural-ODE inference is about 20x faster than the Dopri5 numerical solver (0.008 s vs 0.135 s).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use a neural-SDE asset-price predictor instead of an LSTM | DE-OP for dynamic portfolio optimization · Dynamic portfolio optimization, n=50 test | optimality gap: 21.17% -> 9.11% | Helped |
| Use a neural-SDE asset-price predictor instead of a Feed Forward predictor | DE-OP for dynamic portfolio optimization · Dynamic portfolio optimization, n=50 test | optimality gap: 102.45% -> 9.11% | Helped |
| Omit asset-price dynamics and use a static proxy optimizer | Dynamic portfolio optimization · Dynamic portfolio optimization, n=50 test | optimality gap: exceeding 100% | Hurt |
| Add generator dynamics and stability constraints to the optimization model | AC-OPF proxy optimization · IEEE 57-bus | stability violation: 23.30 ± 0.206 -> 0.00 | Helped |
| Add generator dynamics and stability constraints to the optimization model | AC-OPF proxy optimization · IEEE 57-bus | stability violation: 23.10 ± 0.219 -> 0.00 | Helped |
| Add generator dynamics and stability constraints to the optimization model | AC-OPF proxy optimization · IEEE 57-bus | stability violation: 28.60 ± 0.232 -> 0.00 | Helped |
| Replace Dopri5 numerical solving with neural-ODE inference | Generator dynamics prediction · Dopri5 generator dynamics | inference/computational time (sec): 0.135±0.015 -> 0.008±0 | Helped |
| Replace Bosh3 numerical solving with neural-ODE inference | Generator dynamics prediction · Bosh3 generator dynamics | inference/computational time (sec): 0.446\pm0.039 -> 0.017\pm0 | Helped |
| Use DE-OP with generator dynamics rather than a steady-state proxy | IEEE 57-bus AC-OPF · IEEE 57-bus | optimality gap (at steady-state): 0.16 ± 0.01 -> 0.22 ± 0.02 | Hurt |
Surprising findings
- Static proxy optimizers that ignore dynamics have optimality gaps over 100% on the portfolio task, more than twice the dynamic solution's suboptimality.
- DE-OP has slightly higher steady-state optimality gap (about 0.5% higher) than the best baseline, which the authors attribute to the restricted stability-constrained feasible space.
Limitations
- Steady-state optimality gap is not a valid measure of the stability-constrained objective; exact optimal solutions for the dynamic problem are intractable, so comparisons are indirect.
- Evaluation limited to two power systems and a simplified portfolio model; results depend on the chosen neural-DE surrogate quality.
Experimental setup
Datasets: WSCC 9-bus IEEE 57-bus Nasdaq asset prices synthetic SDE trajectories
Benchmarks: WSCC 9-bus stability-constrained AC-OPF IEEE 57-bus stability-constrained AC-OPF dynamic portfolio optimization (n=50)
Metrics: stability violation flow violation boundary violation optimality gap inference time l2 error of state estimates
Baselines: LD DC3 PDL MSE PINN LSTM Feed Forward network
Builds on: Lagrangian Dual Learning Primal-Dual Learning (PDL) DC3 neural ODE neural SDE
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