We propose a new data-driven approach for learning the fundamental solutions (Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions. Building off the theory of functional linear regression (FLR), we estimate the best-fit Green's function and bias term of the fundamental solution in a reproducing kernel Hilbert space (RKHS) which allows us to regularize their smoothness and impose various structural constraints. We derive a general representer theorem for operator RKHSs to approximate the original infinite-dimensional regression problem by a finite-dimensional one, reducing the search space to a parametric class of Green's functions. In order to study the prediction error of our Green's function estimator, we extend prior results on FLR with scalar outputs to the case with functional outputs. Finally, we demonstrate our method on several linear PDEs including the Poisson, Helmholtz, Schr\"{o}dinger, Fokker-Planck, and heat equation. We highlight its robustness to noise as well as its ability to generalize to new data with varying degrees of smoothness and mesh discretization without any additional training.
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One of the primary reasons behind the success of neural networks has been the emergence of an array of new, highly-successful optimizers, perhaps most importantly the Adam optimizer. It is wiedely used for training neural networks, yet notoriously hard to interpret. Lacking a clear physical intuition, Adam is difficult to generalize to manifolds. Some attempts have been made to directly apply parts of the Adam algorithm to manifolds or to find an underlying structure, but a full generalization has remained elusive. In this work a new approach is presented that leverages the special structure of the manifolds which are relevant for optimization of neural networks, such as the Stiefel manifold, the symplectic Stiefel manifold, the Grassmann manifold and the symplectic Grassmann manifold: all of these are homogeneous spaces and as such admit a global tangent space representation. This global tangent space representation is used to perform all of the steps in the Adam optimizer. The resulting algorithm is then applied to train a transformer for which orthogonality constraints are enforced up to machine precision and we observe significant speed-ups in the training process. Optimization of neural networks where they weights do not lie on a manifold is identified as a special case of the presented framkework. This allows for a flexible implementation in which the learning rate is adapted simultaneously for all parameters, irrespective of whether they are an element of a general manifold or a vector space.
We consider a supervised learning setup in which the goal is to predicts an outcome from a sample of irregularly sampled time series using Neural Controlled Differential Equations (Kidger, Morrill, et al. 2020). In our framework, the time series is a discretization of an unobserved continuous path, and the outcome depends on this path through a controlled differential equation with unknown vector field. Learning with discrete data thus induces a discretization bias, which we precisely quantify. Using theoretical results on the continuity of the flow of controlled differential equations, we show that the approximation bias is directly related to the approximation error of a Lipschitz function defining the generative model by a shallow neural network. By combining these result with recent work linking the Lipschitz constant of neural networks to their generalization capacities, we upper bound the generalization gap between the expected loss attained by the empirical risk minimizer and the expected loss of the true predictor.
Partial differential equations (PDEs) that fit scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects, such as physics and finance. The data-driven discovery of PDEs from scientific data thrives as a new attempt to model complex phenomena in nature, but the effectiveness of current practice is typically limited by the scarcity of data and the complexity of phenomena. Especially, the discovery of PDEs with highly nonlinear coefficients from low-quality data remains largely under-addressed. To deal with this challenge, we propose a novel physics-guided learning method, which can not only encode observation knowledge such as initial and boundary conditions but also incorporate the basic physical principles and laws to guide the model optimization. We theoretically show that our proposed method strictly reduces the coefficient estimation error of existing baselines, and is also robust against noise. Extensive experiments show that the proposed method is more robust against data noise, and can reduce the estimation error by a large margin. Moreover, all the PDEs in the experiments are correctly discovered, and for the first time we are able to discover three-dimensional PDEs with highly nonlinear coefficients.
The generalization of neural networks is a central challenge in machine learning, especially concerning the performance under distributions that differ from training ones. Current methods, mainly based on the data-driven paradigm such as data augmentation, adversarial training, and noise injection, may encounter limited generalization due to model non-smoothness. In this paper, we propose to investigate generalization from a Partial Differential Equation (PDE) perspective, aiming to enhance it directly through the underlying function of neural networks, rather than focusing on adjusting input data. Specifically, we first establish the connection between neural network generalization and the smoothness of the solution to a specific PDE, namely ``transport equation''. Building upon this, we propose a general framework that introduces adaptive distributional diffusion into transport equation to enhance the smoothness of its solution, thereby improving generalization. In the context of neural networks, we put this theoretical framework into practice as PDE+ (\textbf{PDE} with \textbf{A}daptive \textbf{D}istributional \textbf{D}iffusion) which diffuses each sample into a distribution covering semantically similar inputs. This enables better coverage of potentially unobserved distributions in training, thus improving generalization beyond merely data-driven methods. The effectiveness of PDE+ is validated in extensive settings, including clean samples and various corruptions, demonstrating its superior performance compared to SOTA methods.
This work introduces an empirical quadrature-based hyperreduction procedure and greedy training algorithm to effectively reduce the computational cost of solving convection-dominated problems with limited training. The proposed approach circumvents the slowly decaying $n$-width limitation of linear model reduction techniques applied to convection-dominated problems by using a nonlinear approximation manifold systematically defined by composing a low-dimensional affine space with bijections of the underlying domain. The reduced-order model is defined as the solution of a residual minimization problem over the nonlinear manifold. An online-efficient method is obtained by using empirical quadrature to approximate the optimality system such that it can be solved with mesh-independent operations. The proposed reduced-order model is trained using a greedy procedure to systematically sample the parameter domain. The effectiveness of the proposed approach is demonstrated on two shock-dominated computational fluid dynamics benchmarks.
We propose a deep importance sampling method that is suitable for estimating rare event probabilities in high-dimensional problems. We approximate the optimal importance distribution in a general importance sampling problem as the pushforward of a reference distribution under a composition of order-preserving transformations, in which each transformation is formed by a squared tensor-train decomposition. The squared tensor-train decomposition provides a scalable ansatz for building order-preserving high-dimensional transformations via density approximations. The use of composition of maps moving along a sequence of bridging densities alleviates the difficulty of directly approximating concentrated density functions. To compute expectations over unnormalized probability distributions, we design a ratio estimator that estimates the normalizing constant using a separate importance distribution, again constructed via a composition of transformations in tensor-train format. This offers better theoretical variance reduction compared with self-normalized importance sampling, and thus opens the door to efficient computation of rare event probabilities in Bayesian inference problems. Numerical experiments on problems constrained by differential equations show little to no increase in the computational complexity with the event probability going to zero, and allow to compute hitherto unattainable estimates of rare event probabilities for complex, high-dimensional posterior densities.
We show that most structured prediction problems can be solved in linear time and space by considering them as partial orderings of the tokens in the input string. Our method computes real numbers for each token in an input string and sorts the tokens accordingly, resulting in as few as 2 total orders of the tokens in the string. Each total order possesses a set of edges oriented from smaller to greater tokens. The intersection of total orders results in a partial order over the set of input tokens, which is then decoded into a directed graph representing the desired structure. Experiments show that our method achieves 95.4 LAS and 96.9 UAS by using an intersection of 2 total orders, 95.7 LAS and 97.1 UAS with 4 on the English Penn Treebank dependency parsing benchmark. Our method is also the first linear-complexity coreference resolution model and achieves 79.2 F1 on the English OntoNotes benchmark, which is comparable with state of the art.
This paper deals with the problem of efficient sampling from a stochastic differential equation, given the drift function and the diffusion matrix. The proposed approach leverages a recent model for probabilities \cite{rudi2021psd} (the positive semi-definite -- PSD model) from which it is possible to obtain independent and identically distributed (i.i.d.) samples at precision $\varepsilon$ with a cost that is $m^2 d \log(1/\varepsilon)$ where $m$ is the dimension of the model, $d$ the dimension of the space. The proposed approach consists in: first, computing the PSD model that satisfies the Fokker-Planck equation (or its fractional variant) associated with the SDE, up to error $\varepsilon$, and then sampling from the resulting PSD model. Assuming some regularity of the Fokker-Planck solution (i.e. $\beta$-times differentiability plus some geometric condition on its zeros) We obtain an algorithm that: (a) in the preparatory phase obtains a PSD model with L2 distance $\varepsilon$ from the solution of the equation, with a model of dimension $m = \varepsilon^{-(d+1)/(\beta-2s)} (\log(1/\varepsilon))^{d+1}$ where $1/2\leq s\leq1$ is the fractional power to the Laplacian, and total computational complexity of $O(m^{3.5} \log(1/\varepsilon))$ and then (b) for Fokker-Planck equation, it is able to produce i.i.d.\ samples with error $\varepsilon$ in Wasserstein-1 distance, with a cost that is $O(d \varepsilon^{-2(d+1)/\beta-2} \log(1/\varepsilon)^{2d+3})$ per sample. This means that, if the probability associated with the SDE is somewhat regular, i.e. $\beta \geq 4d+2$, then the algorithm requires $O(\varepsilon^{-0.88} \log(1/\varepsilon)^{4.5d})$ in the preparatory phase, and $O(\varepsilon^{-1/2}\log(1/\varepsilon)^{2d+2})$ for each sample. Our results suggest that as the true solution gets smoother, we can circumvent the curse of dimensionality without requiring any sort of convexity.
We develop a class of data-driven generative models that approximate the solution operator for parameter-dependent partial differential equations (PDE). We propose a novel probabilistic formulation of the operator learning problem based on recently developed generative denoising diffusion probabilistic models (DDPM) in order to learn the input-to-output mapping between problem parameters and solutions of the PDE. To achieve this goal we modify DDPM to supervised learning in which the solution operator for the PDE is represented by a class of conditional distributions. The probabilistic formulation combined with DDPM allows for an automatic quantification of confidence intervals for the learned solutions. Furthermore, the framework is directly applicable for learning from a noisy data set. We compare computational performance of the developed method with the Fourier Network Operators (FNO). Our results show that our method achieves comparable accuracy and recovers the noise magnitude when applied to data sets with outputs corrupted by additive noise.
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
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