Motivation of the study. Many real-world phenomena produce time series representing the evolution of physical states, examples include the measurement of neuronal activity in neuroscience, the tracking of mechanical systems in engineering, and the monitoring of ecological processes.
However, these data are often noisy, incomplete, or measured at different sample rates, making it challenging to reconstruct the true underlying signals. Standard smoothing can help, but neglecting known physics may lead to inaccurate results, especially with poor or sparse data. In contrast, physics-informed methods embed mechanistic insights into the smoothing framework, leveraging prior knowledge to guide state reconstruction.
Beyond state trajectories reconstruction, model identification – ****estimating a system’s parametric structure – is of great interest in many applications. In this setting, the same framework can be employed to fit an assumed parametric dynamical model to the available data, providing deeper insights into the system under study.
Objective. The main objective of this thesis is to develop a physics-informed smoothing and model identification framework for dynamical systems, with a focus on robustness to noise and missing data. Specifically, this thesis will:
- Possibly Extend the Approach from Linear to Nonlinear Systems
- Develop a dynamical-informed differential penalty for time series smoothing
- Implement Model Identification procedures
- Handle Missing Data and Inhomogeneous Sampling
Contacts: Pietro Donelli, Marta Gandolla
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Things to know (if you agree to start the thesis)
Prerequisites
To undertake this thesis, you should have:
- Attended a courses in Numerical Analysis or Numerical methods (such as Metodi Analitici e Numerici per l’ingegneria)
- Experience in C++ coding
- (Preferred) Attended courses in Systems Theory or similar
- MOTIVATION!
Logistics & Collaboration
- The methodology will be implemented into the C++ template library fdaPDE
- Code Management: All code will be shared and versioned via GitHub (see [Git Seminar by Giacomo] for guidance).
- Thesis Writing: The thesis must be written using Overleaf.
- Shared Folder: You’ll be granted access to a shared folder where all relevant materials (literature, code, data) must be organized. This will serve as your final delivery package before graduation.
- Team Meetings: You will be included in BioMecc team meetings, where we discuss experimental protocols and ongoing projects with the lab.
Phases of the work
PHASE 1 | Introduction
- Study the provided materials and conduct a literature review.
- Identify and propose promising methodologies for implementation
Milestones
M1: Comprehensive review of existing approaches and proposal of a refined method. Contributions and novel ideas are encouraged! Jointly define the approach(es) to be implemented!
PHASE 2 | Physics-Informed Smoothing for Dynamical systems
- Extend an existing smoothing framework for vector-valued functions by incorporating a penalty term that encodes the known differential equations governing the system.
- Conduct a simulation study to validate its performance in a controlled setting.
- Compare the proposed methodology with other approaches available in the literature.
- Ensure that this physics-informed penalty leverages prior knowledge to guide the smoothing process, mitigating the adverse effects of noise and sparse measurements.
M2: An efficient and tested (on synthetic data) method for Physics-Informed Smoothing for Dynamical systems.
PHASE 3 | Model Identification
- Start with an assumed structure (e.g., linear or low-order nonlinear) of the dynamical system.
- Derive and implement parameter estimation techniques that exploit the observed state trajectories, ensuring that the inferred parameters respect the underlying physics.
M3: Model identification procedure for tuning the parameters of the assumed dynamical model.
PHASE 4 | Missing Data
- Develop strategies to handle real-world complications such as intermittent sensor failures, varying sampling rates across different state variables, and other forms of missingness.
- Integrate these strategies seamlessly into the proposed smoothing and parameter identification framework.
M4: Established method for handling missing data.
PHASE 5 | Application
- Many real-world phenomena produce time series representing the evolution of physical states, examples include the measurement of neuronal activity in neuroscience, the tracking of mechanical systems in engineering, and the monitoring of ecological processes.
M5: Application of the proposed methodology to a real-word application.
PHASE 6 | Writing and presentation
- Agreement on the index and title of the thesis
- Share the writing plan
- Write the thesis
- Presentation preparation and discussion with the team
- Preparation of a short video describing objectives and obtained results of your thesis
M6: Thesis upload; M7: Thesis video; M8: Graduation