Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving Partial Differential Equations (PDEs) by incorporating physical constraints into deep learning models. However, standard PINNs often require a large number of training samples to achieve high accuracy, leading to increased computational costs. To address this issue, we propose Active Learning-Driven PINNs (AL-PINN), which integrates Uncertainty Quantification (UQ) and Active Learning (AL) strategies to optimize sample selection dynamically. AL-PINN utilizes Monte Carlo Dropout to estimate epistemic uncertainty in the model predictions, enabling the adaptive selection of high-uncertainty regions for additional training. This approach significantly enhances learning efficiency by focusing computational resources on the most informative data points. We evaluate AL-PINN on benchmark PDE problems with known analytical solutions and real-world WeatherBench climate data. Our results demonstrate that AL-PINN achieves comparable or superior accuracy compared to traditional PINNs while reducing the number of required training samples. The proposed framework is particularly beneficial for scientific and engineering applications where data collection is expensive or limited, such as climate modeling, medical simulations, and material science. Our findings highlight the potential of active learning in accelerating PINN-based PDE solvers while maintaining high accuracy and computational efficiency.

Intraventricular vector flow mapping (iVFM) seeks to enhance and quantify color Doppler in cardiac imaging. In this study, we propose novel alternatives to the traditional iVFM optimization scheme by utilizing physics-informed neural networks (PINNs) and a physics-guided nnU-Net-based supervised approach. When evaluated on simulated color Doppler images derived from a patient-specific computational fluid dynamics model and in vivo Doppler acquisitions, both approaches demonstrate comparable reconstruction performance to the original iVFM algorithm. The efficiency of PINNs is boosted through dual-stage optimization and pre-optimized weights. On the other hand, the nnU-Net method excels in generalizability and real-time capabilities. Notably, nnU-Net shows superior robustness on sparse and truncated Doppler data while maintaining independence from explicit boundary conditions. Overall, our results highlight the effectiveness of these methods in reconstructing intraventricular vector blood flow. The study also suggests potential applications of PINNs in ultrafast color Doppler imaging and the incorporation of fluid dynamics equations to derive biomarkers for cardiovascular diseases based on blood flow.

Physics-Informed Neural Networks (PINNs) have become a prominent application of deep learning in scientific computation, as they are powerful approximators of solutions to nonlinear partial differential equations (PDEs). There have been numerous attempts to facilitate the training process of PINNs by adjusting the weight of each component of the loss function, called adaptive loss-balancing algorithms. In this paper, we propose an Augmented Lagrangian relaxation method for PINNs (AL-PINNs). We treat the initial and boundary conditions as constraints for the optimization problem of the PDE residual. By employing Augmented Lagrangian relaxation, the constrained optimization problem becomes a sequential max-min problem so that the learnable parameters $\lambda$ adaptively balance each loss component. Our theoretical analysis reveals that the sequence of minimizers of the proposed loss functions converges to an actual solution for the Helmholtz, viscous Burgers, and Klein--Gordon equations. We demonstrate through various numerical experiments that AL-PINNs yield a much smaller relative error compared with that of state-of-the-art adaptive loss-balancing algorithms.