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Characterization of Thermal Systems from Noisy and Low-resolution Measurements Using Dynamic Mode Decomposition
M. E. P. Silva, L. S. Araujo, F. T. Colombo, A. Cunha, S. da Silva
TL;DR
Thermal dynamics are difficult to identify from sparse, noisy, and low-resolution measurements. This chapter applies Dynamic Mode Decomposition with preprocessing and rank selection to two thermal-data scenarios, finding that truncation recovers dominant behavior while balancing stability, detail, and noise sensitivity.
Problem
Identifying thermal dynamics from incomplete, noisy, and low-resolution measurements remains challenging because these observations obscure heat-transfer structures.
Method
The chapter evaluates DMD with preprocessing, controlled truncation, and mode interpretation across sparse thermocouple measurements and degraded thermal images.
Results
DMD recovers coherent thermal structures in both scenarios when rank is carefully selected, with low ranks simplifying spatial variation and higher ranks increasing noise sensitivity.
Takeaways & Limitations
Rank selection is a modeling parameter that determines the balance between stable dominant-behavior descriptions and detailed but noise-sensitive reconstructions.
Takeaways & Limitations
The analysis is limited to two representative experimental configurations and standard DMD with controlled truncation.
Abstract
from arXiv · showhide
Thermal monitoring in practical applications is often constrained by sparse sensing, measurement noise, and limited spatial resolution, which hinder the identification of heat transfer dynamics. In such settings, calibrating high-fidelity physical models is computationally demanding, motivating data-driven approaches. Dynamic Mode Decomposition (DMD) provides a framework for extracting spatiotemporal structures from measurement data, but its standard formulation is sensitive to noise and degraded observations. This chapter examines the use of DMD under these constraints, focusing on preprocessing and truncation strategies that affect stability and interpretability. Two cases are considered: forced convection with thermocouple data and transient heat conduction from degraded thermal images. The number of retained modes is treated as a modeling parameter that governs the trade-off between reconstruction fidelity and noise sensitivity. The results indicate that DMD recovers dominant thermal behavior from both sparse and degraded datasets when the truncation level is appropriately selected. Low-rank models provide stable but simplified descriptions, while higher-rank models improve spatial detail at the cost of increased noise sensitivity.
1 Introduction
The chapter examines DMD as a data-driven tool for extracting dominant thermal dynamics from sparse, noisy, and low-resolution measurements. It shows that preprocessing and rank selection govern the balance between stable simplification, spatial detail, and noise sensitivity.
- Motivation: Industrial thermal monitoring is constrained by limited sensing, measurement noise, and reduced spatial resolution, making accurate first-principles modeling difficult.These constraints occur in applications including thermal energy conversion, refrigeration, and chemical processing.
- Methodological context: DMD identifies spatiotemporal structures from time-resolved data through linear representations derived from snapshot sequences.The method is rooted in Koopman operator theory and has been applied to heat transfer and other fields.
- Contribution: The chapter adapts DMD through preprocessing, rank selection, and mode interpretation rather than introducing a fundamentally new algorithm.These choices are evaluated for sparse sensors and degraded thermal images to improve reliability and extract dominant thermal dynamics.
- Study cases: Two cases evaluate DMD under distinct measurement modalities: noisy thermocouple data from forced convection and degraded thermal images of transient heat conduction in a metallic plate.The image data are intentionally degraded through noise addition and resolution reduction.
- Key findings: DMD recovers coherent spatiotemporal structures from sparse and degraded data when truncation is carefully selected.Low-rank models capture dominant behavior but may oversimplify spatial variations, whereas higher-rank models improve detail while increasing noise sensitivity.
- Practical significance: DMD provides direct measured-data insight into interactions among temperature, flow, and heat transfer when detailed physical models are unavailable or difficult to calibrate.Its usefulness remains limited by measurement quality.
2 Problem Statement and Hypotheses
This section frames thermal-dynamics identification from incomplete, noisy measurements and proposes DMD-based reduced-order modeling directly from observables. It hypothesizes that coherent low-dimensional modes remain identifiable and that truncation rank controls the trade-off between reconstruction fidelity and noise sensitivity.
- Problem formulation: Thermal-dynamics identification is challenged by sparse sensors, limited spatial resolution, and measurement noise that obscure heat-transfer spatiotemporal structures.These limitations arise in both sensor-based and imaging measurements.
- Problem formulation: The measurement model separates information loss from dimensionality reduction through C and additive noise contamination through η(t).C represents sparse sensor placement or spatial averaging, while η(t) denotes measurement noise.
- DMD formulation: The objective is to identify a linear operator A from time-resolved observables and use its dynamic modes to reconstruct and interpret thermal-field evolution.The reduced-order representation is constructed without access to the full state x(t).
- DMD formulation: Standard DMD can yield perturbation-sensitive or physically ambiguous modes under noisy, low-resolution measurements, with preprocessing and truncation rank governing noise amplification.These choices determine the balance between capturing relevant dynamics and amplifying noise.
- Hypotheses and cases: The hypotheses test whether a small coherent-mode set captures dominant variance and whether truncation rank trades reconstruction fidelity against noise sensitivity across forced-convection and transient-conduction cases.The cases use sparse sensor data and degraded thermal images, with results evaluated through normalized reconstruction errors across truncation levels.
3 DMD for Noisy and Low-Resolution Thermal Data
This section presents a practical DMD workflow for noisy and low-resolution thermal data, emphasizing preprocessing and truncated SVD to improve decomposition stability and interpretability. The truncation rank controls the trade-off between reconstruction detail and robustness to noise.
- Practical workflow: The workflow combines dimensionality reduction, controlled rank selection, and preprocessing to extract coherent spatiotemporal patterns from sparse, noisy, or degraded thermal datasets.The section identifies forced convection and transient heat conduction as the experimental settings examined next.
- DMD formulation: DMD represents measured thermal observables as time-indexed snapshots, with sensor data collecting thermocouple readings and thermal images vectorized from pixel arrays.For thermal images, a p × q array produces m = pq measurements.
- Dimensionality reduction: Direct least-squares estimation can overfit noisy, sparsely sampled data, so truncated SVD provides a reduced-order representation based on the r dominant singular values and vectors.The reduced operator is constructed from the truncated SVD components before extracting DMD eigenvalues and modes.
- Mode reconstruction: DMD eigenvalues and modes encode temporal evolution and spatial structure, enabling reconstruction of the thermal field as a compact superposition of modes.This reconstruction provides a compact description of the underlying dynamics.
- Rank selection: Low truncation ranks filter noise and stabilize representations but may suppress spatial variability and transients, whereas higher ranks increase reconstruction detail while risking noise sensitivity and spurious oscillations.Rank selection is therefore treated as a modeling parameter balancing reconstruction fidelity and robustness.
4 Results and Discussion
The results show that DMD can recover dominant thermal behavior from sparse sensors and degraded thermal images, with truncation rank controlling the balance between coherent structure, reconstruction fidelity, and noise sensitivity. Low ranks provide stable but simplified descriptions, whereas higher ranks add detail but increasingly risk fitting measurement noise.
- Forced convection: In the forced-convection case, r = 1 captures the global axial heating gradient but imposes identical temporal behavior across all sensors.This misses the differential response between the upstream fan-side sensor and downstream heater-side sensor during transients.
- Forced convection: Using r = 2 introduces a sign-changing spatial mode that captures convective asymmetry and distinct transient trajectories between sensor locations.The two-mode reconstruction represents the lag between upstream cooling and downstream heating and recovers the measured signals’ qualitative non-uniform response.
- Forced convection: With r = 3, the reconstruction closely follows all measured signals, but the third mode represents finer, lower-amplitude variations and cannot be separated from noise by truncation alone.The three modes form a hierarchy of global gradient, primary convective asymmetry, and finer variations requiring physical interpretation.
- Transient heat conduction: For transient heat conduction, r = 1 preserves the broad heated-zone shape but assigns nearly identical spatial distributions across time steps.The dominant mode reflects the lowest spatial frequency and diffusive heat spread from the source.
- Transient heat conduction: Increasing to r = 10 recovers finer, physically meaningful spatial structures, including heat-peak broadening, cooling, asymmetric thermal-front spread, and edge cooling.The reconstructed conduction and cooling dynamics match the main spatiotemporal features of the original images more faithfully than the single-mode case.
- Cross-case interpretation: At r = 40, visual reconstruction fidelity improves further, but modes beyond approximately index 20 show irregular high-frequency oscillations consistent with noise fitting.Across both cases, the first ten modes capture primary thermal structures, while higher truncation levels become increasingly noise-influenced.
5 Final Remarks
DMD recovers dominant thermal dynamics from sparse measurements and degraded thermal images when truncation is appropriately selected. Truncation rank controls the trade-off between reconstruction fidelity, spatial detail, and noise sensitivity.
- Results: DMD recovers dominant dynamical behavior in forced convection using only a few temperature measurements.Low-order representations capture the global thermal trend stably, while higher-order models improve fidelity and introduce greater spatial variability.
- Results: DMD reconstructs main spatiotemporal temperature structures from degraded thermal images despite noise contamination and reduced resolution.Low-rank models produce smooth dominant heat distributions, whereas higher-rank models recover finer spatial features with increased noise sensitivity.
- Results: Truncation rank acts as a modeling parameter governing the trade-off between reconstruction fidelity and robustness across sparse measurements and degraded images.Appropriate rank selection extracts coherent thermal structures without requiring explicit knowledge of the governing equations.
- Implications: DMD provides a practical reduced-order modeling approach for thermo-fluid systems when detailed physical models are unavailable or difficult to calibrate.Dimensionality reduction, modal decomposition, and preprocessing organize thermal-dynamics analysis directly from measurements while remaining sensitive to data-quality limitations.
- Limitations and Future Work: The study is limited to two representative experimental configurations and standard DMD with controlled truncation.Future developments include advanced DMD variants, systematic rank-selection strategies, physics-informed constraints, and noise-robust methods.