A hybrid Kalman filtering and proper generalized decomposition algorithmfor real-time identification of partial differential governing equationsystems
Bibliographic record
Abstract
A novel combination of Kalman filtering (KF) and proper generalized decomposition (PGD) is introduced in this study for parameter estimation in systems with partial differential equations (PDE). The literature reveals that the KF develops a physicsinformed digital twin of engineering systems using its first-order transition and measurement functions for system identification purposes. The transition and measurement functions predict states and output in each iteration, and the rest of the KF algorithm updates the predicted states based on the actual measurements. However, in many cases, it is very complicated to decouple and discretize the partial differential governing equations to derive the transition and measurement functions. Also, finding an explicit expression representative of the relationship between the desired parameters and states is another obstacle in derivation of the transition function in the joint state-parameter estimation problem. Similarly for the measurement function, since most of the time the PDE has no closed-form solution, deriving an unambiguous expression for the measurement function is not straightforward. To overcome the above-mentioned drawbacks related to implementing the KF for systems with numerous DoFs and PDEs, we propose to use the PGD instead of the transition and measurement functions for state propagation in each time step within the identification process. The PGD performs an offline parametric solution with all possible scenarios in a predefined range and provides a library of solutions for all the desired parameters and states. The KF picks one response from the PGD library based on the estimated state value and desired parameters from the previous time step and predicts the desired state vector for the next iteration in a real-time scheme. The PGD helps to avoid the repetitive and extensive computation of the transition and measurement functions in each iteration. To show the proficiency of the method, in addition to a simulation study, we develop a digital twin for a lab-scale cantilever beam with a nonlinear spring at the tip and track the change of stiffness of the nonlinear spring over time. The results show that the PGD-KF could be used for damage identification of systems with high DoFs and partial differential governing equations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".