Description:The present invention provides a novel system and method for predictive modeling of semantic drift in natural language using Koopman operator theory, a mathematical framework that enables the transformation of non-linear dynamical systems into linear systems in an elevated function space. This invention addresses key limitations in existing NLP-based semantic drift models, such as the inability to capture non-linear dynamics, high retraining costs, and limited predictive capacity.
Core Principle:
The invention is based on the observation that semantic drift—the change in word meaning over time—can be modeled as a non-linear dynamical system. Traditional models attempt to learn this change through embedding retraining, which is computationally intensive and historically bound. By contrast, this invention applies the Koopman operator to lift word embeddings into a higher-dimensional space where their temporal evolution can be linearly approximated and predicted.
System Architecture:
The invention comprises the following main modules:
1. Temporal Alignment Module:
a. Aligns word embeddings across different time slices to ensure temporal consistency.
b. Handles preprocessing of diachronic corpora to extract time-specific embeddings using models like Word2Vec or BERT.
2. Polynomial Observable Mapping:
a. Transforms input word embeddings into a higher-dimensional function space using polynomial observables (typically of degree 2), enabling the modeling of non-linear dynamics.
3. Koopman Operator Computation:
a. Uses pairs of temporally-aligned embeddings to compute the Koopman operator using least-squares fitting and pseudo-inverse matrix operations.
b. Enables representation of temporal evolution via a linear operator in the transformed space.
4. Predictive Modeling Module:
a. Applies the computed Koopman operator to the transformed initial embeddings to forecast future embeddings over multiple time steps.
5. Eigenvalue Analysis Module:
a. Performs Koopman eigenvalue decomposition to identify:
(i) Stable components (|?| ˜ 1)
(ii) Oscillatory components (? with non-zero imaginary parts)
(iii) Unstable/drifting components (|?| > 1)
b. Enables semantic interpretability of drift behavior.

6. Visualization Interface:

a. Maps predicted and actual embeddings using PCA or t-SNE for interpretability.
b. Generates time-series plots of cosine similarity and error metrics.
Algorithms:
Algorithm 1 (Koopman Operator Computation):
a. Constructs two embedding matrices from sequential time steps.
b. Applies polynomial transformation to the earlier matrix.
c. Computes Koopman operator via least-squares optimization.
Algorithm 2 (Semantic Forecasting):
a. Transforms current embedding into observable space.
b. Repeatedly applies Koopman operator to simulate future states.
c. Maps predicted observables back to the embedding space.
Technical Advantages:
Predictive Capability: Enables long-term forecasting of word meanings without continual model retraining.
Computational Efficiency: Reduces the need for complex temporal models by replacing them with matrix operations.
Interpretability: Decomposes semantic drift into interpretable eigenmodes (stable, oscillatory, or transient).
Generalizability: Can be applied to multiple embedding types (static or contextual) and languages.

Example Use-Case:
For the word “guys”, known to have undergone semantic broadening over decades, the model successfully predicts future embeddings consistent with modern usage. Cosine similarity and mean squared error metrics validate the model’s predictive accuracy against baseline and dynamic retraining methods.
Industrial Application:
This invention can be implemented in AI-driven search engines, chatbots, semantic surveillance systems, and historical language processing tools. It can also be deployed in real-time applications for monitoring evolving social sentiments, hate speech detection, and dynamic knowledge graphs.
, Claims:1. A system for predictive modeling of semantic drift, comprising:
a. A temporal alignment module for normalizing word embeddings across multiple time periods;
b. A Koopman operator computation unit that transforms non-linear semantic trajectories into a linear system using polynomial observables.;
c. A predictive module for evolving transformed embeddings linearly over time using the Koopman operator;
d. An eigenvalue analysis module for categorizing semantic stability, oscillatory behavior, and rapid shifts;

2. A method for predicting long-term semantic drift, comprising:

a. Loading and aligning word embeddings for temporal consistency;
b. Computing a Koopman operator to represent semantic drift as a linear dynamical system, and transient fluctuations using Koopman Mode Decomposition.
c. Decomposing semantic evolution into stable trends, oscillatory modes;
d. Predicting future word meanings by linearly evolving transformed states in the Koopman space

3. The system of claim 1, wherein the Koopman operator is formulated using polynomial observables of degree 2, enhancing computational efficiency.
4. The method of claim 2, wherein the system employs Koopman eigenvalue analysis to:
a. Distinguish between stable, oscillatory, and rapidly shifting word meanings;
b. Provide long-term forecasts of semantic evolution;

5. The system of claim 1, further comprising a visualization module that illustrates temporal shifts using PCA-reduced embedding trajectories, highlighting directional movements over time.

6. The method of claim 2, wherein the system is configured to:
a. Perform predictive modeling without requiring continuous retraining;
b. Compare predictive accuracy against existing models, including Temporal Word2Vec and BERT-based diachronic embeddings.