Mathematical Structure for Biomedicine: Variational Models, PDEs, and Statistical Learning

Dear Colleagues,

This Special Issue gathers work where mathematical structure—variational principles, PDEs, and inverse problems—guides data-driven inference in biomedicine. We welcome contributions that unite model-based methods with statistical learning (e.g., gradient boosting, GAMs, kernel methods), modern operator learning, and interpretable predictive modelling for images and signals/time series (e.g., MRI/CT/US/Echo, ECG/EEG, EHR). Priority will be given to studies that provide theoretical insight (well-posedness, stability, error bounds) and reliable inference (uncertainty, calibration, shift robustness, decision utility) with reproducible implementations. We especially encourage cross-modal studies that bridge imaging and signals and include external validation on independent cohorts. Benchmark and software papers with open datasets/simulators and transparent protocols are welcome, as are tutorials that clarify mathematical foundations for practitioners. Submissions should make explicit how mathematics informs model design and evaluation (e.g., ablations isolating the role of priors/constraints) and report decision-relevant metrics alongside accuracy. Carefully argued negative or neutral results that sharpen understanding of when structure helps, or fails, are also in scope.

Topics of interest include, but are not limited to, the following:

• Variational/PDE formulations for imaging: reconstruction, denoising, inpainting, segmentation, and registration;
• Inverse problems with learned priors; plug-and-play/RED; unrolled optimization; and hybrid solvers;
• Statistical learning for biomedical prediction: gradient boosting, GAMs, sparse/regularized models, and kernels;
• Operator learning (e.g., DeepONet, FNO) for physiological forward/adjoint models and constitutive laws;
• Physics-/PDE-informed neural models; structure-aware diffusion/score priors; and constrained generation
• Geometry/topology-aware methods; graphs/manifolds; and harmonic/graph PDEs in biomedicine;
• Multimodal fusion of images, signals, and tabular clinical data; joint reconstruction–segmentation–prediction;
• Reliability: uncertainty quantification, calibration, drift detection, out-of-distribution robustness, and decision-curve analysis;
• Benchmarks, simulators, datasets, and open software with mathematical fidelity and reproducibility.

Dr. Andrej Novak
Prof. Dr. Davor Horvatic
Guest Editors

Manuscript Submission Information

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• variational methods
• partial differential equations
• image processing and analysis
• statistical learning
• operator learning
• biomedical imaging
• biomedical signals and time series
• uncertainty quantification
• calibration and robustness
• physics-informed modelling