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Background: Perfusion magnetic resonance imaging (MRI)s plays a central role in the diagnosis and monitoring of neurovascular or neurooncological disease. However, conventional processing techniques are limited in their ability to capture relevant characteristics of the perfusion dynamics and suffer from a lack of standardization. Purpose: We propose a physics‐informed deep learning framework which is capable of analyzing dynamic susceptibility contrast perfusion MRI data and recovering the dynamic tissue response with high accuracy. Methods: The framework uses physics‐informed neural networks (PINNs) to learn the voxel‐wise TRF, which represents the dynamic response of the local vascular network to the contrast agent bolus. The network output is stabilized by total variation and elastic net regularization. Parameter maps of normalized cerebral blood flow (nCBF) and volume (nCBV) are then calculated from the predicted residue functions. The results are validated using extensive comparisons to values derived by conventional Tikhonov‐regularized singular value decomposition (TiSVD), in silico simulations and an in vivo dataset of perfusion MRI exams of patients with high‐grade gliomas. Results: The simulation results demonstrate that PINN‐derived residue functions show a high concordance with the true functions and that the calculated values of nCBF and nCBV converge towards the true values for higher contrast‐to‐noise ratios. In the in vivo dataset, we find high correlations between conventionally derived and PINN‐predicted perfusion parameters (Pearson's rho for nCBF: 0.84±0.03$0.84 \pm 0.03$ and nCBV: 0.92±0.03$0.92 \pm 0.03$) and very high indices of image similarity (structural similarity index for nCBF: 0.91±0.03$0.91 \pm 0.03$ and for nCBV: 0.98±0.00$0.98 \pm 0.00$). Conclusions: PINNs can be used to analyze perfusion MRI data and stably recover the response functions of the local vasculature with high accuracy. [ABSTRACT FROM AUTHOR]