In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is discovering unknown physics and the corresponding equations. However, prior to achieving this goal, major challenges remain to be resolved, including learning PDE under noisy data and limited discrete data. To overcome these challenges, in this work, a deep-learning based data-driven method, called DL-PDE, is developed to discover the governing PDEs of underlying physical processes. The DL-PDE method combines deep learning via neural networks and data-driven discovery of PDE via sparse regressions. In the DL-PDE, a neural network is first trained, and then a large amount of meta-data is generated, and the required derivatives are calculated by automatic differentiation. Finally, the form of PDE is discovered by sparse regression. The proposed method is tested with physical processes, governed by groundwater flow equation, convection-diffusion equation, Burgers equation and Korteweg-de Vries (KdV) equation, for proof-of-concept and applications in real-world engineering settings. The proposed method achieves satisfactory results when data are noisy and limited.
Université de Paris（巴黎大学）；INSERM U1284（法国国家健康与医学研究院 U1284）；CRI（计算研究中心）；IBM Research Paris-Saclay（IBM 巴黎-萨克雷研究院）
Automated model discovery of partial differential equations (PDEs) usually considers a single experiment or dataset to infer the underlying governing equations. In practice, experiments have inherent natural variability in parameters, initial and boundary conditions that cannot be simply averaged out. We introduce a randomised adaptive group Lasso sparsity estimator to promote grouped sparsity and implement it in a deep learning based PDE discovery framework. It allows to create a learning bias that implies the a priori assumption that all experiments can be explained by the same underlying PD…
Center for Research and Interdisciplinarity (CRI)（跨学科研究中心（CRI））；Université de Paris, INSERM U1284（巴黎大学，法国国家健康与医学研究院U1284研究所）；F-75006 Paris, France（法国巴黎75006）
Discovering the partial differential equations underlying spatio-temporal datasets from very limited and highly noisy observations is of paramount interest in many scientific fields. However, it remains an open question to know when model discovery algorithms based on sparse regression can actually recover the underlying physical processes. In this work, we show the design matrices used to infer the equations by sparse regression can violate the irrepresentability condition (IRC) of the Lasso, even when derived from analytical PDE solutions (i.e. without additional noise). Sparse regression te…
Peking University（北京大学）；Eastern Institute for Advanced Study（东方高等研究院）；Eastern Institute of Technology（东方理工大学）；National Center for Applied Mathematics Shenzhen（深圳国家应用数学中心）；Southern University of Science and Technology（南方科技大学）
The working mechanisms of complex natural systems tend to abide by concise and profound partial differential equations (PDEs). Methods that directly mine equations from data are called PDE discovery, which reveals consistent physical laws and facilitates our adaptive interaction with the natural world. In this paper, an enhanced deep reinforcement-learning framework is proposed to uncover symbolically concise open-form PDEs with little prior knowledge. Particularly, based on a symbol library of basic operators and operands, a structure-aware recurrent neural network agent is designed and seaml…
Brown University（布朗大学）；Pacific Northwest National Laboratory（西北太平洋国家实验室）
We propose a Bayesian physics-informed neural network (B-PINN) to solve both forward and inverse nonlinear problems described by partial differential equations (PDEs) and noisy data. In this Bayesian framework, the Bayesian neural network (BNN) combined with a PINN for PDEs serves as the prior while the Hamiltonian Monte Carlo (HMC) or the variational inference (VI) could serve as an estimator of the posterior. B-PINNs make use of both physical laws and scattered noisy measurements to provide predictions and quantify the aleatoric uncertainty arising from the noisy data in the Bayesian framewo…
Peking University（北京大学）；Southern University of Science and Technology（南方科技大学）；Peng Cheng Laboratory（鹏城实验室）
Data-driven methods have recently made great progress in the discovery of partial differential equations (PDEs) from spatial-temporal data. However, several challenges remain to be solved, including sparse noisy data, incomplete candidate library, and spatially- or temporally-varying coefficients. In this work, a new framework, which combines neural network, genetic algorithm and adaptive methods, is put forward to address all of these challenges simultaneously. In the framework, a trained neural network is utilized to calculate derivatives and generate a large amount of meta-data, which solve…
Peking University（北京大学）；Southern University of Science and Technology（南方科技大学）；Peng Cheng Laboratory（鹏城实验室）
Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order derivatives, the performance of existing methods is unsatisfactory, especially when the data are sparse and noisy. It is also difficult to discover heterogeneous parametric PDEs where heterogeneous parameters are embedded in the partial differential operators. In this work, a new framework combining deep-learning and integral form is proposed to handle the above-men…
We prove the support recovery for a general class of linear and nonlinear evolutionary partial differential equation (PDE) identification from a single noisy trajectory using $\ell_1$ regularized Pseudo-Least Squares model~($\ell_1$-PsLS). In any associative $\mathbb{R}$-algebra generated by finitely many differentiation operators that contain the unknown PDE operator, applying $\ell_1$-PsLS to a given data set yields a family of candidate models with coefficients $\mathbf{c}(λ)$ parameterized by the regularization weight $λ\geq 0$. The trace of $\{\mathbf{c}(λ)\}_{λ\geq 0}$ suffers from high …
We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network, and extracts the PDE by equating derivatives of the neural network approximation. Our method applies to PDEs which are linear combinations of user-defined dictionary functions, and generalizes previous methods that only consider parabolic PDEs. We introduce a regularization scheme that prevents the function approximation from overfitting the data and forces i…
Huazhong University of Science and Technology（华中科技大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）；Georgia Institute of Technology（佐治亚理工学院）；Wuhan University（武汉大学）；University of Cambridge（剑桥大学）；University of Luxembourg（卢森堡大学）；Weill Cornell Medical College（威尔康奈尔医学院）
The study presents a general framework for discovering underlying Partial Differential Equations (PDEs) using measured spatiotemporal data. The method, called Sparse Spatiotemporal System Discovery ($\text{S}^3\text{d}$), decides which physical terms are necessary and which can be removed (because they are physically negligible in the sense that they do not affect the dynamics too much) from a pool of candidate functions. The method is built on the recent development of Sparse Bayesian Learning; which enforces the sparsity in the to-be-identified PDEs, and therefore can balance the model compl…
We introduce DeepMoD, a Deep learning based Model Discovery algorithm. DeepMoD discovers the partial differential equation underlying a spatio-temporal data set using sparse regression on a library of possible functions and their derivatives. A neural network approximates the data and constructs the function library, but it also performs the sparse regression. This construction makes it extremely robust to noise, applicable to small data sets, and, contrary to other deep learning methods, does not require a training set. We benchmark our approach on several physical problems such as the Burger…
The behavior of many dynamical systems follow complex, yet still unknown partial differential equations (PDEs). While several machine learning methods have been proposed to learn PDEs directly from data, previous methods are limited to discrete-time approximations or make the limiting assumption of the observations arriving at regular grids. We propose a general continuous-time differential model for dynamical systems whose governing equations are parameterized by message passing graph neural networks. The model admits arbitrary space and time discretizations, which removes constraints on the …
Qu & Co B.V.（Qu & Co 公司）；Czech Technical University（捷克技术大学）；University of Exeter（埃克塞特大学）
Quantum computing promises to speed up some of the most challenging problems in science and engineering. Quantum algorithms have been proposed showing theoretical advantages in applications ranging from chemistry to logistics optimization. Many problems appearing in science and engineering can be rewritten as a set of differential equations. Quantum algorithms for solving differential equations have shown a provable advantage in the fault-tolerant quantum computing regime, where deep and wide quantum circuits can be used to solve large linear systems like partial differential equations (PDEs) …
Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the system evolution at much coarser, meso- or macroscopic length scales. Discovering those coarse-grained effective PDEs can lead to considerable savings in computation-intensive tasks like prediction or control. We propose a framework combining artificial neural networks with multiscale computation, in the form of equation-free numerics, for efficient discovery of such macro-sca…
School of Mathematical Sciences（数学科学学院）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；Peking University, Beijing, China（北京大学北京中国）；Beijing Computational Science Research Center（北京计算科学研究中心）；Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Center for Data Science, Peking University（北京大学数据科学中心）；Beijing Institute of Big Data Research（北京大数据研究院）
