TY - JOUR AB - Scientific machine learning is a new class of approaches that integrate physical knowledge and mechanistic models with data-driven techniques to uncover the governing equations of complex processes. Among the available approaches, universal differential equations (UDEs) combine prior knowledge in the form of mechanistic formulations with universal function approximators, such as neural networks. Integral to the efficacy of UDEs is the joint estimation of parameters for both the mechanistic formulations and the universal function approximators using empirical data. However, the robustness and applicability of these resultant models hinge upon the rigorous quantification of uncertainties associated with their parameters and predictive capabilities. In this work, we provide a formalization of uncertainty quantification (UQ) for UDEs and investigate key frequentist and Bayesian methods. By analyzing three synthetic examples of varying complexity, we evaluate the validity and efficiency of ensembles, variational inference and Markov-chain Monte Carlo sampling as epistemic UQ methods for UDEs.This article is part of the theme issue 'Uncertainty quantification for healthcare and biological systems (Part 2)'. AU - Schmid, N.* AU - Fernandes del Pozo, D.* AU - Waegeman, W.* AU - Hasenauer, J. C1 - 73972 C2 - 57253 CY - 6-9 Carlton House Terrace, London Sw1y 5ag, England TI - Assessment of uncertainty quantification in universal differential equations. JO - Philos. Trans. R. Soc. A - Math. Phys. Eng. Sci. VL - 383 IS - 2293 PB - Royal Soc PY - 2025 SN - 1364-503X ER - TY - JOUR AB - A key feature of optoacoustic imaging is the ability to illuminate tissue at multiple wavelengths and therefore record images with a spectral dimension. While optoacoustic images at single wavelengths reveal morphological features, in analogy to ultrasound imaging or X-ray imaging, spectral imaging concedes sensing of intrinsic chromophores and externally administered agents that can reveal physiological, cellular and subcellular functions. Nevertheless, identification of spectral moieties within images obtained at multiple wavelengths requires spectral unmixing techniques, which present a unique mathematical problem given the three-dimensional nature of the optoacoustic images. Herein we discuss progress with spectral unmixing techniques developed for multispectral optoacoustic tomography. We explain how different techniques are required for accurate sensing of intrinsic tissue chromophores such as oxygenated and deoxygenated haemoglobin versus extrinsically administered photo-absorbing agents and nanoparticles. Finally, we review recent developments that allow accurate quantification of blood oxygen saturation (sO2) by transforming and solving the sO2 estimation problem from the spatial to the spectral domain.This article is part of the themed issue 'Challenges for chemistry in molecular imaging'. AU - Tzoumas, S.* AU - Ntziachristos, V. C1 - 52173 C2 - 43748 CY - London TI - Spectral unmixing techniques for optoacoustic imaging of tissue pathophysiology. JO - Philos. Trans. R. Soc. A - Math. Phys. Eng. Sci. VL - 375 IS - 2107 PB - Royal Soc PY - 2017 SN - 1364-503X ER - TY - JOUR AB - Increasingly complex applications involve large datasets in combination with nonlinear and high dimensional mathematical models. In this context, statistical inference is a challenging issue that calls for pragmatic approaches that take advantage of both Bayesian and frequentist methods. The elegance of Bayesian methodology is founded in the propagation of information content provided by experimental data and prior assumptions to the posterior probability distribution of model predictions. However, for complex applications experimental data and prior assumptions potentially constrain the posterior probability distribution insuciently. In these situations Bayesian Markov chain Monte Carlo sampling can be infeasible. From a frequentist point of view insucient experimental data and prior assumptions can be interpreted as non-identi ability. The pro le likelihood approach o ers to detect and to resolve non-identi ability by experimental design iteratively. Therefore, it allows one to better constrain the posterior probability distribution until Markov chain Monte Carlo sampling can be used securely. Using an application from cell biology we compare both methods and show that a successive application of both methods facilitates a realistic assessment of uncertainty in model predictions. AU - Raue, A.* AU - Kreutz, C.* AU - Theis, F.J. AU - Timmer, J. C1 - 10687 C2 - 30324 TI - Joining forces of Bayesian and frequentist methodology: A study for inference in the presence of non-identiability. JO - Philos. Trans. R. Soc. A - Math. Phys. Eng. Sci. VL - 371 IS - 1984 PB - Royal Society PY - 2013 SN - 1364-503X ER - TY - JOUR AB - Macroscopic optical imaging has rather humble technical origins; it has been mostly implemented by photographic means using appropriate filters, a light source and a camera yielding images of tissues. This approach relates to human vision and perception, and is simple to implement and use. Therefore, it has found wide acceptance, especially in recording fluorescence and bioluminescence signals. Yet, the difficulty in resolving depth and the dependence of the light intensity recorded on tissue optical properties may compromise the accuracy of the approach. Recently, optical technology has seen significant advances that bring a new performance level in optical investigations. Quantitative real-time multi-spectral optical and optoacoustic (photoacoustic) methods enable high-resolution quantitative imaging of tissue and disease biomarkers and can significantly enhance medical vision in diagnostic or interventional procedures such as dermatology, endoscopy, surgery, and various vascular and intravascular imaging applications. This performance is showcased herein and examples are given to illustrate how it is possible to shift the paradigm of optical clinical translation. AU - Ntziachristos, V. C1 - 6029 C2 - 29217 SP - 4666-4678 TI - Clinical translation of optical and optoacoustic imaging. JO - Philos. Trans. R. Soc. A - Math. Phys. Eng. Sci. VL - 369 IS - 1955 PB - Royal Society PY - 2011 SN - 1364-503X ER - TY - JOUR AB - Major field experiment programmes from 1957 projected to 1987 on the disposal of low-level and high-level radioactive waste in salt formations, are briefly described. They include the USA and Germany (Asse salt mine). In 1987 an experiment at the last-named will involve the introduction of 30 vitrified blocks of high-level radwaste with a disposal time of five years. They will contain 7 M Ci137Cs and 3 M Ci90Sr. This experiment will also examine gas production by radiolysis. Dispersal of radwaste into salt formations can be realized and long-term safety proved.-R.K.H. AU - Kuhn, K. C1 - 42289 C2 - 0 SP - 157-161 TI - Field experiments in salt formations. JO - Philos. Trans. R. Soc. A - Math. Phys. Eng. Sci. VL - 319 IS - 1545 PY - 1986 SN - 1364-503X ER -