CASCADE Singularity Proof
21-Simulation Results
Proving CASCADE advantage on physics problems with REAL singularities
13
CASCADE Wins
61.9%
Win Rate
93.4%
Best Improvement
21
Simulations
The Key Insight
CASCADE enables access to "danger zones" near singularities where standard MOO fails due to numerical overflow/divergence. The NNC transform with optimal k regularizes these singularities, allowing exploration of solution regions that baseline cannot reach.
D*[1/r] = -1
Bigeometric derivative is constant
D*[ln(r)] = 1
No divergence near singularity
k = -1.0 optimal
For power-law singularities
Top Improvements
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Singularity Types and Optimal k
| Singularity | Form | Optimal k | Physics Examples |
|---|---|---|---|
| 1/r | Inverse | -1 | Coulomb potential, gravity, vortex cores |
| 1/r^2 | Inverse square | -1 | Gravity force, radiation intensity |
| 1/sqrt(r) | Square root | -0.5 | Crack tip stress, fracture mechanics |
| ln(r) | Logarithmic | -1 | Well pressure, 2D potential |
| 1/(r-r_s) | Horizon | -1 | Black hole horizon, event horizon |
| 1/r^12 | Lennard-Jones | -1 | Molecular repulsion |
Methodology
Test Procedure:
- Create singularity problem for each physics domain
- Run Baseline NSGA2 (50 gen x 50 pop = 2500 evals)
- Run CASCADE with optimal k (same budget)
- Measure: min distance to singularity, solutions in danger zone
Winner Criteria:
- CASCADE gets >10% closer to singularity
- OR CASCADE finds more solutions in danger zone
- OR CASCADE has >1% better hypervolume