Combinatorial Multivector Fields for Real-Time Dynamic System Modeling
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Solution Overview
Problem
Existing methods for processing sampled vector fields in dynamic systems face challenges such as numerical inaccuracies, computational complexity, and reliance on continuous models, which are inefficient and costly, especially for high-dimensional or irregularly sampled data.
Innovation Solution
A combinatorial method that transforms sampled vector fields into discrete directed graphs using integer programming, generating combinatorial multivector fields to model dynamic systems, preserving topological and dynamical structure without continuous models, and enabling efficient, incremental updates.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If conventional continuous models and numerical methods are used to process sampled vector fields, then modeling capability is provided, but computational complexity and cost increase significantly
Solution Approach 1:
The patent replaces continuous mathematical models and numerical computation methods with a discrete combinatorial approach using directed graphs and multivector fields. Instead of solving differential equations or performing intensive numerical analysis, the system uses graph-theoretic operations and combinatorial topology to model dynamic systems, thereby substituting a computationally heavy mechanical/numerical system with a more efficient discrete mathematical system.
Solution Approach 2:
The patent transforms the continuous parameters of traditional modeling into discrete combinatorial parameters. By representing the state space as a directed graph with discrete cells and using combinatorial multivector fields instead of continuous vector fields, the system changes the fundamental parameter type from continuous real-valued parameters to discrete combinatorial structures, reducing computational complexity while preserving essential system behavior.
2Productivity
If discrete or irregularly sampled data is processed using traditional methods, then data processing is achieved, but topological structure and dynamical structure are not preserved
Solution Approach 1:
The patent segments the continuous state space into discrete cells, forming a directed graph representation. By dividing the phase space into manageable discrete units and defining transitions between them, the system can process irregularly sampled data while maintaining the underlying topological structure through the graph's connectivity patterns.
Solution Approach 2:
The patent introduces combinatorial multivector fields as an intermediary structure between the discrete graph representation and the continuous dynamical system behavior. These multivector fields serve as a bridge that preserves topological and dynamical information, allowing the discrete data structure to accurately represent continuous system properties without requiring continuous models.
3Reliability
If continuous models are used for dynamic system modeling, then system behavior can be analyzed, but numerical inaccuracies and computational errors are introduced
Solution Approach 1:
The patent substitutes continuous mathematical models that require numerical approximation with a discrete combinatorial model based on directed graphs and multivector fields. This replacement eliminates the need for numerical solvers and iterative methods, thereby removing the source of numerical inaccuracies and computational errors while maintaining the ability to analyze system behavior through graph-theoretic and topological methods.
Data Source
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AI summary
A computer-implemented method for performing real-time measurements in a technical system, the method comprising: receiving input data representing sampled vector fields (SVF) obtained from real-time sensor measurements of said technical system, wherein said sampled vector fields comprise vectors associated with discrete measurement points within a physical space of said technical system; transforming the input data into a directed graph; generating candidate multivectors from said directed graph; determining an optimal decomposition of a finite topological space into multivectors selected from candidate multivectors; creating a transition graph by augmenting the directed graph with additional intra-multivector edges; and outputting data representing the transition graph as a computer-readable representation of the real-time dynamic behavior of the technical system, wherein said transition graph explicitly models transitions between system states derived directly from the sampled vector fields.