Neural operators (e.g., Fourier Neural Operator, DeepONet) learn mappings between function spaces and increasingly replace expensive numerical simulations. In practice, however, the real process always deviates from the simulation, and real measurements are scarce, noisy, and often only partially available. Within this project, we investigate how an operator pretrained on simulation data can be corrected using few, partially covered observations of the real system (residual or delta modelling, sim-to-real transfer). This thesis lays the methodological foundation on public benchmark problems of operator learning.
First, a neural operator is to be trained on an established benchmark problem (e.g., Darcy flow). Subsequently, a synthetic sim-to-real gap is to be created, for instance through perturbed coefficients or additional terms in the reference model. The core of the thesis is the implementation and comparison of methods for residual correction of the pretrained operator from few observations, where observations are only partially covered due to masking. The relationship between observation coverage, number of measurements, noise level, and achievable correction quality is to be quantified systematically. As a baseline, at least Gaussian process regression is to be included. The resulting evaluation environment should be built for reuse.
Solid Python skills and experience with a deep learning framework (PyTorch preferred); fundamentals of machine learning; interest in numerical simulation and scientific computing. Prior knowledge of neural operators or Gaussian processes is helpful but not required.
Sebastian Baum
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In deep drawing of cylindrical parts, periodic shape deviations known as earing form at the rim. In a production line, the rim contour h(φ) can often only be captured on partial arcs, and depending on the measurement position, the angular orientation of the part relative to the tool is unknown. Within this project, this poses a well-defined estimation problem: how well can a periodic contour be reconstructed from partial, noisy observations, and under which conditions can the unknown rotation be estimated from the characteristic, unequal expression of the ears themselves?
Based on synthetic contour data (parametric test contours and contours from preliminary finite element runs), an estimation method for reconstructing h(φ) from partial arcs is to be implemented and its conditioning analysed as a function of coverage. Subsequently, registration of the unknown rotation via matching against the asymmetric earing signature is to be implemented (e.g., cross-correlation), and its success probability quantified systematically as a function of asymmetry level, noise level, and coverage. The results feed directly into the selection of the measurement position in the project. Optionally (research project), the rotation can alternatively be treated probabilistically as a latent variable.
Solid Python skills (NumPy/SciPy); fundamentals of signal processing or estimation theory (Fourier analysis, regression); a structured, mathematically rigorous way of working. Knowledge of forming technology is not required.
Sebastian Baum
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