Neurophysiology Model (10D)
[MAL-Seeker Execution Record] High-Dimensional Dense System: Neurophysiology Model
Variables: 10-Dimensional System (Dense Matrix)
Search Bounds: $[-10.0, 10.0]$
1. Problem Definition (Challenging High-Dimensional Dense Systems)
This analysis focuses on a 10-variable system of nonlinear equations modeling a neurophysiological network.
The most defining feature of this system is that each equation $f_i$ depends on all other variables $x_j$. This results in a "Dense Matrix" where every element of the Jacobian is non-zero, causing computational loads to skyrocket as dimensionality increases. It is a challenging problem that demands resilience against the "curse of dimensionality" and heavy matrix operations.
2. Execution Results
Final log of the all-solution search using MAL-Seeker (Project Bellatrix V1.0.0) in a 10D space.
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🏆 NEUROPHYSIOLOGY MODEL (10D) 🏆
Found 3 Unique Roots (Dense Matrix System)
Execution Time: 124.058014 seconds
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Root 1: ( 0.0000000, 0.0000000, ..., 0.0000000)
Root 2: (-2.5413813, -2.5413813, ..., -2.5413813)
Root 3: ( 3.5413813, 3.5413813, ..., 3.5413813)The real roots were identified from a vast 10-dimensional space with pinpoint accuracy. The entire search and convergence process was completed in 124 seconds.
3. Mathematical Verification: Consistency with Theoretical Values
Based on the symmetry of the problem, assuming all variables are equal ( $x_1 = \dots = x_{10} = x$ ) simplifies the system to the following analytical form:
$x^3 - x^2 - 9x = 0$
$\Rightarrow x = 0, \quad x = \frac{1 \pm \sqrt{37}}{2} (\approx -2.54138126, \,\, 3.54138126)$
The numerical solutions output by MAL-Seeker match the theoretical values **perfectly up to 7 decimal places**, proving the extremely high reliability of our engine in high-dimensional calculations.
I am highly satisfied with the result of extracting all three real roots without a single omission in a 10-dimensional dense matrix system where all variables are intricately linked.
Standard solvers often succumb to the "curse of dimensionality," resulting in computational failure or getting lost in the sea of local minima. The fact that our engine completed the intense spatial scan and matrix operations without crashing—directly hitting the theoretical solutions—demonstrates that Bellatrix possesses sufficient robustness and practicality for advanced nonlinear modeling in R&D environments.
4. Analysis Environment
- Analysis Engine: MAL-Seeker (Project Bellatrix V1.0.0 / C)
- PC: MacBook
- OS: macOS Monterey 12.7.6
- CPU: 1.2GHz Dual-Core Intel Core m5
- Memory: 8GB
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