The use of neural networks for solving differential equations is practically difficult due to the exponentially increasing runtime of autodifferentiation when computing high-order derivatives. We propose $n$-TangentProp, the natural extension of the TangentProp formalism \cite{simard1991tangent} to arbitrarily many derivatives. $n$-TangentProp computes the exact derivative $d^n/dx^n f(x)$ in quasilinear, instead of exponential time, for a densely connected, feed-forward neural network $f$ with a smooth, parameter-free activation function. We validate our algorithm empirically across a range of depths, widths, and number of derivatives. We demonstrate that our method is particularly beneficial in the context of physics-informed neural networks where \ntp allows for significantly faster training times than previous methods and has favorable scaling with respect to both model size and loss-function complexity as measured by the number of required derivatives. The code for this paper can be found at https://github.com/kyrochi/n\_tangentprop.

Mesh smoothing methods can enhance mesh quality by eliminating distorted elements, leading to improved convergence in simulations. To balance the efficiency and robustness of traditional mesh smoothing process, previous approaches have employed supervised learning and reinforcement learning to train intelligent smoothing models. However, these methods heavily rely on labeled dataset or prior knowledge to guide the models' learning. Furthermore, their limited capacity to enhance mesh connectivity often restricts the effectiveness of smoothing. In this paper, we first systematically analyze the learning mechanisms of recent intelligent smoothing methods and propose a prior-free reinforcement learning model for intelligent mesh smoothing. Our proposed model integrates graph neural networks with reinforcement learning to implement an intelligent node smoothing agent and introduces, for the first time, a mesh connectivity improvement agent. We formalize mesh optimization as a Markov Decision Process and successfully train both agents using Twin Delayed Deep Deterministic Policy Gradient and Double Dueling Deep Q-Network in the absence of any prior data or knowledge. We verified the proposed model on both 2D and 3D meshes. Experimental results demonstrate that our model achieves feature-preserving smoothing on complex 3D surface meshes. It also achieves state-of-the-art results among intelligent smoothing methods on 2D meshes and is 7.16 times faster than traditional optimization-based smoothing methods. Moreover, the connectivity improvement agent can effectively enhance the quality distribution of the mesh.

Machine learning (ML) models are nowadays used in complex applications in various domains, such as medicine, bioinformatics, and other sciences. Due to their black box nature, however, it may sometimes be hard to understand and trust the results they provide. This has increased the demand for reliable visualization tools related to enhancing trust in ML models, which has become a prominent topic of research in the visualization community over the past decades. To provide an overview and present the frontiers of current research on the topic, we present a State-of-the-Art Report (STAR) on enhancing trust in ML models with the use of interactive visualization. We define and describe the background of the topic, introduce a categorization for visualization techniques that aim to accomplish this goal, and discuss insights and opportunities for future research directions. Among our contributions is a categorization of trust against different facets of interactive ML, expanded and improved from previous research. Our results are investigated from different analytical perspectives: (a) providing a statistical overview, (b) summarizing key findings, (c) performing topic analyses, and (d) exploring the data sets used in the individual papers, all with the support of an interactive web-based survey browser. We intend this survey to be beneficial for visualization researchers whose interests involve making ML models more trustworthy, as well as researchers and practitioners from other disciplines in their search for effective visualization techniques suitable for solving their tasks with confidence and conveying meaning to their data.