Understanding the properties of the stochastic phase field models is crucial to model processes in several practical applications, such as soft matters and phase separation in random environments. To describe such random evolution, this work proposes and studies two mathematical models and their numerical approximations for parabolic stochastic partial differential equation (SPDE) with a logarithmic Flory--Huggins energy potential. These multiscale models are built based on a regularized energy technique and thus avoid possible singularities of coefficients. According to the large deviation principle, we show that the limit of the proposed models with small noise naturally recovers the classical dynamics in deterministic case. Moreover, when the driving noise is multiplicative, the Stampacchia maximum principle holds which indicates the robustness of the proposed model. One of the main advantages of the proposed models is that they can admit the energy evolution law and asymptotically preserve the Stampacchia maximum bound of the original problem. To numerically capture these asymptotic behaviors, we investigate the semi-implicit discretizations for regularized logrithmic SPDEs. Several numerical results are presented to verify our theoretical findings.
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Gradient clipping is a popular modification to standard (stochastic) gradient descent, at every iteration limiting the gradient norm to a certain value $c >0$. It is widely used for example for stabilizing the training of deep learning models (Goodfellow et al., 2016), or for enforcing differential privacy (Abadi et al., 2016). Despite popularity and simplicity of the clipping mechanism, its convergence guarantees often require specific values of $c$ and strong noise assumptions. In this paper, we give convergence guarantees that show precise dependence on arbitrary clipping thresholds $c$ and show that our guarantees are tight with both deterministic and stochastic gradients. In particular, we show that (i) for deterministic gradient descent, the clipping threshold only affects the higher-order terms of convergence, (ii) in the stochastic setting convergence to the true optimum cannot be guaranteed under the standard noise assumption, even under arbitrary small step-sizes. We give matching upper and lower bounds for convergence of the gradient norm when running clipped SGD, and illustrate these results with experiments.
State-of-the-art machine-learning-based models are a popular choice for modeling and forecasting energy behavior in buildings because given enough data, they are good at finding spatiotemporal patterns and structures even in scenarios where the complexity prohibits analytical descriptions. However, their architecture typically does not hold physical correspondence to mechanistic structures linked with governing physical phenomena. As a result, their ability to successfully generalize for unobserved timesteps depends on the representativeness of the dynamics underlying the observed system in the data, which is difficult to guarantee in real-world engineering problems such as control and energy management in digital twins. In response, we present a framework that combines lumped-parameter models in the form of linear time-invariant (LTI) state-space models (SSMs) with unsupervised reduced-order modeling in a subspace-based domain adaptation (SDA) framework. SDA is a type of transfer-learning (TL) technique, typically adopted for exploiting labeled data from one domain to predict in a different but related target domain for which labeled data is limited. We introduce a novel SDA approach where instead of labeled data, we leverage the geometric structure of the LTI SSM governed by well-known heat transfer ordinary differential equations to forecast for unobserved timesteps beyond observed measurement data. Fundamentally, our approach geometrically aligns the physics-derived and data-derived embedded subspaces closer together. In this initial exploration, we evaluate the physics-based SDA framework on a demonstrative heat conduction scenario by varying the thermophysical properties of the source and target systems to demonstrate the transferability of mechanistic models from a physics-based domain to a data domain.
Neural networks are the state-of-the-art for many approximation tasks in high-dimensional spaces, as supported by an abundance of experimental evidence. However, we still need a solid theoretical understanding of what they can approximate and, more importantly, at what cost and accuracy. One network architecture of practical use, especially for approximation tasks involving images, is convolutional (residual) networks. However, due to the locality of the linear operators involved in these networks, their analysis is more complicated than for generic fully connected neural networks. This paper focuses on sequence approximation tasks, where a matrix or a higher-order tensor represents each observation. We show that when approximating sequences arising from space-time discretisations of PDEs we may use relatively small networks. We constructively derive these results by exploiting connections between discrete convolution and finite difference operators. Throughout, we design our network architecture to, while having guarantees, be similar to those typically adopted in practice for sequence approximation tasks. Our theoretical results are supported by numerical experiments which simulate linear advection, the heat equation, and the Fisher equation. The implementation used is available at the repository associated to the paper.
The Immersed Boundary (IB) method of Peskin (J. Comput. Phys., 1977) is useful for problems involving fluid-structure interactions or complex geometries. By making use of a regular Cartesian grid that is independent of the geometry, the IB framework yields a robust numerical scheme that can efficiently handle immersed deformable structures. Additionally, the IB method has been adapted to problems with prescribed motion and other PDEs with given boundary data. IB methods for these problems traditionally involve penalty forces which only approximately satisfy boundary conditions, or they are formulated as constraint problems. In the latter approach, one must find the unknown forces by solving an equation that corresponds to a poorly conditioned first-kind integral equation. This operation can require a large number of iterations of a Krylov method, and since a time-dependent problem requires this solve at each time step, this method can be prohibitively inefficient without preconditioning. In this work, we introduce a new, well-conditioned IB formulation for boundary value problems, which we call the Immersed Boundary Double Layer (IBDL) method. We present the method as it applies to Poisson and Helmholtz problems to demonstrate its efficiency over the original constraint method. In this double layer formulation, the equation for the unknown boundary distribution corresponds to a well-conditioned second-kind integral equation that can be solved efficiently with a small number of iterations of a Krylov method. Furthermore, the iteration count is independent of both the mesh size and immersed boundary point spacing. The method converges away from the boundary, and when combined with a local interpolation, it converges in the entire PDE domain. Additionally, while the original constraint method applies only to Dirichlet problems, the IBDL formulation can also be used for Neumann conditions.
Current ethical debates on the use of artificial intelligence (AI) in health care treat AI as a product of technology in three ways: First, by assessing risks and potential benefits of currently developed AI-enabled products with ethical checklists; second, by proposing ex ante lists of ethical values seen as relevant for the design and development of assisting technology, and third, by promoting AI technology to use moral reasoning as part of the automation process. Subsequently, we propose a fourth approach to AI, namely as a methodological tool to assist ethical reflection. We provide a concept of an AI-simulation informed by three separate elements: 1) stochastic human behavior models based on behavioral data for simulating realistic settings, 2) qualitative empirical data on value statements regarding internal policy, and 3) visualization components that aid in understanding the impact of changes in these variables. The potential of this approach is to inform an interdisciplinary field about anticipated ethical challenges or ethical trade-offs in concrete settings and, hence, to spark a re-evaluation of design and implementation plans. This may be particularly useful for applications that deal with extremely complex values and behavior or with limitations on the communication resources of affected persons (e.g., persons with dementia care or for care of persons with cognitive impairment). Simulation does not replace ethical reflection but does allow for detailed, context-sensitive analysis during the design process and prior to implementation. Finally, we discuss the inherently quantitative methods of analysis afforded by stochastic simulations as well as the potential for ethical discussions and how simulations with AI can improve traditional forms of thought experiments and future-oriented technology assessment.
Unlike conventional grid and mesh based methods for solving partial differential equations (PDEs), neural networks have the potential to break the curse of dimensionality, providing approximate solutions to problems where using classical solvers is difficult or impossible. While global minimization of the PDE residual over the network parameters works well for boundary value problems, catastrophic forgetting impairs the applicability of this approach to initial value problems (IVPs). In an alternative local-in-time approach, the optimization problem can be converted into an ordinary differential equation (ODE) on the network parameters and the solution propagated forward in time; however, we demonstrate that current methods based on this approach suffer from two key issues. First, following the ODE produces an uncontrolled growth in the conditioning of the problem, ultimately leading to unacceptably large numerical errors. Second, as the ODE methods scale cubically with the number of model parameters, they are restricted to small neural networks, significantly limiting their ability to represent intricate PDE initial conditions and solutions. Building on these insights, we develop Neural IVP, an ODE based IVP solver which prevents the network from getting ill-conditioned and runs in time linear in the number of parameters, enabling us to evolve the dynamics of challenging PDEs with neural networks.
In this paper, we propose a novel numerical scheme to optimize the gradient flows for learning energy-based models (EBMs). From a perspective of physical simulation, we redefine the problem of approximating the gradient flow utilizing optimal transport (i.e. Wasserstein) metric. In EBMs, the learning process of stepwise sampling and estimating data distribution performs the functional gradient of minimizing the global relative entropy between the current and target real distribution, which can be treated as dynamic particles moving from disorder to target manifold. Previous learning schemes mainly minimize the entropy concerning the consecutive time KL divergence in each learning step. However, they are prone to being stuck in the local KL divergence by projecting non-smooth information within smooth manifold, which is against the optimal transport principle. To solve this problem, we derive a second-order Wasserstein gradient flow of the global relative entropy from Fokker-Planck equation. Compared with existing schemes, Wasserstein gradient flow is a smoother and near-optimal numerical scheme to approximate real data densities. We also derive this near-proximal scheme and provide its numerical computation equations. Our extensive experiments demonstrate the practical superiority and potentials of our proposed scheme on fitting complex distributions and generating high-quality, high-dimensional data with neural EBMs.
The locally modified finite element method, which is introduced in [Frei, Richter: SINUM 52(2014), p. 2315-2334], is a simple fitted finite element method that is able to resolve weak discontinuities in interface problems. The method is based on a fixed structured coarse mesh, which is then refined into sub-elements to resolve an interior interface. In this work, we extend the locally modified finite element method {in two space dimensions} to second order using an isoparametric approach in the interface elements. Thereby we need to take care that the resulting curved edges do not lead to degenerate sub-elements. We prove optimal a priori error estimates in the $L^2$-norm and in a discrete energy norm. Finally, we present numerical examples to substantiate the theoretical findings.
In temporal extensions of Answer Set Programming (ASP) based on linear-time, the behavior of dynamic systems is captured by sequences of states. While this representation reflects their relative order, it abstracts away the specific times associated with each state. However, timing constraints are important in many applications like, for instance, when planning and scheduling go hand in hand. We address this by developing a metric extension of linear-time temporal equilibrium logic, in which temporal operators are constrained by intervals over natural numbers. The resulting Metric Equilibrium Logic provides the foundation of an ASP-based approach for specifying qualitative and quantitative dynamic constraints. To this end, we define a translation of metric formulas into monadic first-order formulas and give a correspondence between their models in Metric Equilibrium Logic and Monadic Quantified Equilibrium Logic, respectively. Interestingly, our translation provides a blue print for implementation in terms of ASP modulo difference constraints.
Autonomous Nano Aerial Vehicles have been increasingly popular in surveillance and monitoring operations due to their efficiency and maneuverability. Once a target location has been reached, drones do not have to remain active during the mission. It is possible for the vehicle to perch and stop its motors in such situations to conserve energy, as well as maintain a static position in unfavorable flying conditions. In the perching target estimation phase, the steady and accuracy of a visual camera with markers is a significant challenge. It is rapidly detectable from afar when using a large marker, but when the drone approaches, it quickly disappears as out of camera view. In this paper, a vision-based target poses estimation method using multiple markers is proposed to deal with the above-mentioned problems. First, a perching target with a small marker inside a larger one is designed to improve detection capability at wide and close ranges. Second, the relative poses of the flying vehicle are calculated from detected markers using a monocular camera. Next, a Kalman filter is applied to provide a more stable and reliable pose estimation, especially when the measurement data is missing due to unexpected reasons. Finally, we introduced an algorithm for merging the poses data from multi markers. The poses are then sent to the position controller to align the drone and the marker's center and steer it to perch on the target. The experimental results demonstrated the effectiveness and feasibility of the adopted approach. The drone can perch successfully onto the center of the markers with the attached 25mm-diameter rounded magnet.
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