We present a novel loss formulation for efficient learning of complex dynamics from governing physics, typically described by partial differential equations (PDEs), using physics-informed neural networks (PINNs). In our experiments, existing versions of PINNs are seen to learn poorly in many problems, especially for complex geometries, as it becomes increasingly difficult to establish appropriate sampling strategy at the near boundary region. Overly dense sampling can adversely impede training convergence if the local gradient behaviors are too complex to be adequately modelled by PINNs. On the other hand, if the samples are too sparse, existing PINNs tend to overfit the near boundary region, leading to incorrect solution. To prevent such issues, we propose a new Boundary Connectivity (BCXN) loss function which provides linear local structure approximation (LSA) to the gradient behaviors at the boundary for PINN. Our BCXN-loss implicitly imposes local structure during training, thus facilitating fast physics-informed learning across entire problem domains with order of magnitude sparser training samples. This LSA-PINN method shows a few orders of magnitude smaller errors than existing methods in terms of the standard L2-norm metric, while using dramatically fewer training samples and iterations. Our proposed LSA-PINN does not pose any requirement on the differentiable property of the networks, and we demonstrate its benefits and ease of implementation on both multi-layer perceptron and convolutional neural network versions as commonly used in current PINN literature.
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In decision-making problems such as the multi-armed bandit, an agent learns sequentially by optimizing a certain feedback. While the mean reward criterion has been extensively studied, other measures that reflect an aversion to adverse outcomes, such as mean-variance or conditional value-at-risk (CVaR), can be of interest for critical applications (healthcare, agriculture). Algorithms have been proposed for such risk-aware measures under bandit feedback without contextual information. In this work, we study contextual bandits where such risk measures can be elicited as linear functions of the contexts through the minimization of a convex loss. A typical example that fits within this framework is the expectile measure, which is obtained as the solution of an asymmetric least-square problem. Using the method of mixtures for supermartingales, we derive confidence sequences for the estimation of such risk measures. We then propose an optimistic UCB algorithm to learn optimal risk-aware actions, with regret guarantees similar to those of generalized linear bandits. This approach requires solving a convex problem at each round of the algorithm, which we can relax by allowing only approximated solution obtained by online gradient descent, at the cost of slightly higher regret. We conclude by evaluating the resulting algorithms on numerical experiments.
Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms and software capabilities for quadratization of non-autonomous ODEs. We provide existence results, depending on the regularity of the input function, for cases when a quadratic-bilinear system can be obtained through quadratization. We further develop existence results and an algorithm that generalizes the process of quadratization for systems with arbitrary dimension that retain the nonlinear structure when the dimension grows. For such systems, we provide dimension-agnostic quadratization. An example is semi-discretized PDEs, where the nonlinear terms remain symbolically identical when the discretization size increases. As an important aspect for practical adoption of this research, we extended the capabilities of the QBee software towards both non-autonomous systems of ODEs and ODEs with arbitrary dimension. We present several examples of ODEs that were previously reported in the literature, and where our new algorithms find quadratized ODE systems with lower dimension than the previously reported lifting transformations. We further highlight an important area of quadratization: reduced-order model learning. This area can benefit significantly from working in the optimal lifting variables, where quadratic models provide a direct parametrization of the model that also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlights these advantages.
This work introduces a reduced order modeling (ROM) framework for the solution of parameterized second-order linear elliptic partial differential equations formulated on unfitted geometries. The goal is to construct efficient projection-based ROMs, which rely on techniques such as the reduced basis method and discrete empirical interpolation. The presence of geometrical parameters in unfitted domain discretizations entails challenges for the application of standard ROMs. Therefore, in this work we propose a methodology based on i) extension of snapshots on the background mesh and ii) localization strategies to decrease the number of reduced basis functions. The method we obtain is computationally efficient and accurate, while it is agnostic with respect to the underlying discretization choice. We test the applicability of the proposed framework with numerical experiments on two model problems, namely the Poisson and linear elasticity problems. In particular, we study several benchmarks formulated on two-dimensional, trimmed domains discretized with splines and we observe a significant reduction of the online computational cost compared to standard ROMs for the same level of accuracy. Moreover, we show the applicability of our methodology to a three-dimensional geometry of a linear elastic problem.
In the task of predicting spatio-temporal fields in environmental science using statistical methods, introducing statistical models inspired by the physics of the underlying phenomena that are numerically efficient is of growing interest. Large space-time datasets call for new numerical methods to efficiently process them. The Stochastic Partial Differential Equation (SPDE) approach has proven to be effective for the estimation and the prediction in a spatial context. We present here the advection-diffusion SPDE with first order derivative in time which defines a large class of nonseparable spatio-temporal models. A Gaussian Markov random field approximation of the solution to the SPDE is built by discretizing the temporal derivative with a finite difference method (implicit Euler) and by solving the spatial SPDE with a finite element method (continuous Galerkin) at each time step. The ''Streamline Diffusion'' stabilization technique is introduced when the advection term dominates the diffusion. Computationally efficient methods are proposed to estimate the parameters of the SPDE and to predict the spatio-temporal field by kriging, as well as to perform conditional simulations. The approach is applied to a solar radiation dataset. Its advantages and limitations are discussed.
The design of complex self-organising systems producing life-like phenomena, such as the open-ended evolution of virtual creatures, is one of the main goals of artificial life. Lenia, a family of cellular automata (CA) generalizing Conway's Game of Life to continuous space, time and states, has attracted a lot of attention because of the wide diversity of self-organizing patterns it can generate. Among those, some spatially localized patterns (SLPs) resemble life-like artificial creatures and display complex behaviors. However, those creatures are found in only a small subspace of the Lenia parameter space and are not trivial to discover, necessitating advanced search algorithms. Furthermore, each of these creatures exist only in worlds governed by specific update rules and thus cannot interact in the same one. This paper proposes as mass-conservative extension of Lenia, called Flow Lenia, that solve both of these issues. We present experiments demonstrating its effectiveness in generating SLPs with complex behaviors and show that the update rule parameters can be optimized to generate SLPs showing behaviors of interest. Finally, we show that Flow Lenia enables the integration of the parameters of the CA update rules within the CA dynamics, making them dynamic and localized, allowing for multi-species simulations, with locally coherent update rules that define properties of the emerging creatures, and that can be mixed with neighbouring rules. We argue that this paves the way for the intrinsic evolution of self-organized artificial life forms within continuous CAs.
This paper focuses on parameter estimation and introduces a new method for lower bounding the Bayesian risk. The method allows for the use of virtually \emph{any} information measure, including R\'enyi's $\alpha$, $\varphi$-Divergences, and Sibson's $\alpha$-Mutual Information. The approach considers divergences as functionals of measures and exploits the duality between spaces of measures and spaces of functions. In particular, we show that one can lower bound the risk with any information measure by upper bounding its dual via Markov's inequality. We are thus able to provide estimator-independent impossibility results thanks to the Data-Processing Inequalities that divergences satisfy. The results are then applied to settings of interest involving both discrete and continuous parameters, including the ``Hide-and-Seek'' problem, and compared to the state-of-the-art techniques. An important observation is that the behaviour of the lower bound in the number of samples is influenced by the choice of the information measure. We leverage this by introducing a new divergence inspired by the ``Hockey-Stick'' Divergence, which is demonstrated empirically to provide the largest lower-bound across all considered settings. If the observations are subject to privatisation, stronger impossibility results can be obtained via Strong Data-Processing Inequalities. The paper also discusses some generalisations and alternative directions.
In this work we develop implicit Active Flux schemes for the scalar advection equation. At every cell interface we approximate the solution by a polynomial in time. This allows to evolve the point values using characteristics and to update the cell averages using fluxes obtained by integrating this polynomial. The resulting schemes have order of convergence up to five, but show almost no oscillations with high frequencies for discontinuous solutions. In numerical experiments we compare the different methods and show an application to network flows.
Detecting sets of relevant patterns from a given dataset is an important challenge in data mining. The relevance of a pattern, also called utility in the literature, is a subjective measure and can be actually assessed from very different points of view. Rule-based languages like Answer Set Programming (ASP) seem well suited for specifying user-provided criteria to assess pattern utility in a form of constraints; moreover, declarativity of ASP allows for a very easy switch between several criteria in order to analyze the dataset from different points of view. In this paper, we make steps toward extending the notion of High Utility Pattern Mining (HUPM); in particular we introduce a new framework that allows for new classes of utility criteria not considered in the previous literature. We also show how recent extensions of ASP with external functions can support a fast and effective encoding and testing of the new framework. To demonstrate the potential of the proposed framework, we exploit it as a building block for the definition of an innovative method for predicting ICU admission for COVID-19 patients. Finally, an extensive experimental activity demonstrates both from a quantitative and a qualitative point of view the effectiveness of the proposed approach. Under consideration in Theory and Practice of Logic Programming (TPLP)
Multimodal learning helps to comprehensively understand the world, by integrating different senses. Accordingly, multiple input modalities are expected to boost model performance, but we actually find that they are not fully exploited even when the multimodal model outperforms its uni-modal counterpart. Specifically, in this paper we point out that existing multimodal discriminative models, in which uniform objective is designed for all modalities, could remain under-optimized uni-modal representations, caused by another dominated modality in some scenarios, e.g., sound in blowing wind event, vision in drawing picture event, etc. To alleviate this optimization imbalance, we propose on-the-fly gradient modulation to adaptively control the optimization of each modality, via monitoring the discrepancy of their contribution towards the learning objective. Further, an extra Gaussian noise that changes dynamically is introduced to avoid possible generalization drop caused by gradient modulation. As a result, we achieve considerable improvement over common fusion methods on different multimodal tasks, and this simple strategy can also boost existing multimodal methods, which illustrates its efficacy and versatility. The source code is available at \url{//github.com/GeWu-Lab/OGM-GE_CVPR2022}.
Unsupervised domain adaptation has recently emerged as an effective paradigm for generalizing deep neural networks to new target domains. However, there is still enormous potential to be tapped to reach the fully supervised performance. In this paper, we present a novel active learning strategy to assist knowledge transfer in the target domain, dubbed active domain adaptation. We start from an observation that energy-based models exhibit free energy biases when training (source) and test (target) data come from different distributions. Inspired by this inherent mechanism, we empirically reveal that a simple yet efficient energy-based sampling strategy sheds light on selecting the most valuable target samples than existing approaches requiring particular architectures or computation of the distances. Our algorithm, Energy-based Active Domain Adaptation (EADA), queries groups of targe data that incorporate both domain characteristic and instance uncertainty into every selection round. Meanwhile, by aligning the free energy of target data compact around the source domain via a regularization term, domain gap can be implicitly diminished. Through extensive experiments, we show that EADA surpasses state-of-the-art methods on well-known challenging benchmarks with substantial improvements, making it a useful option in the open world. Code is available at //github.com/BIT-DA/EADA.
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