arXiv subjects

Justin Sirignano

Publications and source records attributed to Justin Sirignano.

At least 37 records · Page 2Linked to original sources

arXiv2023

Deep Learning Closure of the Navier-Stokes Equations for Transition-Continuum Flows

The predictive accuracy of the Navier-Stokes equations is known to degrade at the limits of the continuum assumption, thereby necessitating expensive and often highly approximate solutions to the Boltzmann equation. While tractable in one spatial dimension, their high dimensionality makes multi-dimensional Boltzmann calculations impractical for all but canonical configurations. It is therefore desirable to augment the Navier-Stokes equations in these regimes. We present an application of a deep learning method to extend the validity of the Navier-Stokes equations to the transition-continuum flows. The technique encodes the missing physics via a neural network, which is trained directly from Boltzmann solutions. While standard DL methods can be considered ad-hoc due to the absence of underlying physical laws, at least in the sense that the systems are not governed by known partial differential equations, the DL framework leverages the a-priori known Boltzmann physics while ensuring that the trained model is consistent with the Navier-Stokes equations. The online training procedure solves adjoint equations, constructed using algorithmic differentiation, which efficiently provide the gradient of the loss function with respect to the learnable parameters. The model is trained and applied to predict stationary, one-dimensional shock thickness in low-pressure argon.

physics.flu-dyn↗

arXiv2023

Neural Q-learning for solving PDEs

Solving high-dimensional partial differential equations (PDEs) is a major challenge in scientific computing. We develop a new numerical method for solving elliptic-type PDEs by adapting the Q-learning algorithm in reinforcement learning. Our "Q-PDE" algorithm is mesh-free and therefore has the potential to overcome the curse of dimensionality. Using a neural tangent kernel (NTK) approach, we prove that the neural network approximator for the PDE solution, trained with the Q-PDE algorithm, converges to the trajectory of an infinite-dimensional ordinary differential equation (ODE) as the number of hidden units $\rightarrow \infty$. For monotone PDE (i.e. those given by monotone operators, which may be nonlinear), despite the lack of a spectral gap in the NTK, we then prove that the limit neural network, which satisfies the infinite-dimensional ODE, converges in $L^2$ to the PDE solution as the training time $\rightarrow \infty$. More generally, we can prove that any fixed point of the wide-network limit for the Q-PDE algorithm is a solution of the PDE (not necessarily under the monotone condition). The numerical performance of the Q-PDE algorithm is studied for several elliptic PDEs.

math.NA↗

arXiv2023

Global Convergence of Deep Galerkin and PINNs Methods for Solving Partial Differential Equations

Numerically solving high-dimensional partial differential equations (PDEs) is a major challenge. Conventional methods, such as finite difference methods, are unable to solve high-dimensional PDEs due to the curse-of-dimensionality. A variety of deep learning methods have been recently developed to try and solve high-dimensional PDEs by approximating the solution using a neural network. In this paper, we prove global convergence for one of the commonly-used deep learning algorithms for solving PDEs, the Deep Galerkin Method (DGM). DGM trains a neural network approximator to solve the PDE using stochastic gradient descent. We prove that, as the number of hidden units in the single-layer network goes to infinity (i.e., in the ``wide network limit"), the trained neural network converges to the solution of an infinite-dimensional linear ordinary differential equation (ODE). The PDE residual of the limiting approximator converges to zero as the training time $\rightarrow \infty$. Under mild assumptions, this convergence also implies that the neural network approximator converges to the solution of the PDE. A closely related class of deep learning methods for PDEs is Physics Informed Neural Networks (PINNs). Using the same mathematical techniques, we can prove a similar global convergence result for the PINN neural network approximators. Both proofs require analyzing a kernel function in the limit ODE governing the evolution of the limit neural network approximator. A key technical challenge is that the kernel function, which is a composition of the PDE operator and the neural tangent kernel (NTK) operator, lacks a spectral gap, therefore requiring a careful analysis of its properties.

math.NA↗

arXiv2023

Dynamic Deep Learning LES Closures: Online Optimization With Embedded DNS

Deep learning (DL) has recently emerged as a candidate for closure modeling of large-eddy simulation (LES) of turbulent flows. High-fidelity training data is typically limited: it is computationally costly (or even impossible) to numerically generate at high Reynolds numbers, while experimental data is also expensive to produce and might only include sparse/aggregate flow measurements. Thus, only a relatively small number of geometries and physical regimes will realistically be included in any training dataset. Limited data can lead to overfitting and therefore inaccurate predictions for geometries and physical regimes that are different from the training cases. We develop a new online training method for deep learning closure models in LES which seeks to address this challenge. The deep learning closure model is dynamically trained during a large-eddy simulation (LES) calculation using embedded direct numerical simulation (DNS) data. That is, in a small subset of the domain, the flow is computed at DNS resolutions in concert with the LES prediction. The closure model then adjusts its approximation to the unclosed terms using data from the embedded DNS. Consequently, the closure model is trained on data from the exact geometry/physical regime of the prediction at hand. An online optimization algorithm is developed to dynamically train the deep learning closure model in the coupled, LES-embedded DNS calculation.

physics.flu-dyn↗

arXiv2022

Deep Learning Closure Models for Large-Eddy Simulation of Flows around Bluff Bodies

A deep learning (DL) closure model for large-eddy simulation (LES) is developed and evaluated for incompressible flows around a rectangular cylinder at moderate Reynolds numbers. Near-wall flow simulation remains a central challenge in aerodynamic modeling: RANS predictions of separated flows are often inaccurate, while LES can require prohibitively small near-wall mesh sizes. The DL-LES model is trained using adjoint PDE optimization methods to match, as closely as possible, direct numerical simulation (DNS) data. It is then evaluated out-of-sample (i.e., for new aspect ratios and Reynolds numbers not included in the training data) and compared against a standard LES model (the dynamic Smagorinsky model). The DL-LES model outperforms dynamic Smagorinsky and is able to achieve accurate LES predictions on a relatively coarse mesh (downsampled from the DNS grid by a factor of four in each Cartesian direction). We study the accuracy of the DL-LES model for predicting the drag coefficient, mean flow, and Reynolds stress. A crucial challenge is that the LES quantities of interest are the steady-state flow statistics; for example, the time-averaged mean velocity $\bar{u}(x) = \displaystyle \lim_{t \rightarrow \infty} \frac{1}{t} \int_0^t u(s,x) ds$. Calculating the steady-state flow statistics therefore requires simulating the DL-LES equations over a large number of flow times through the domain; it is a non-trivial question whether an unsteady partial differential equation model whose functional form is defined by a deep neural network can remain stable and accurate on $t \in [0, \infty)$. Our results demonstrate that the DL-LES model is accurate and stable over large physical time spans, enabling the estimation of the steady-state statistics for the velocity, fluctuations, and drag coefficient of turbulent flows around bluff bodies relevant to aerodynamic applications.

physics.flu-dyn↗

arXiv2022

A Forward Propagation Algorithm for Online Optimization of Nonlinear Stochastic Differential Equations

Optimizing over the stationary distribution of stochastic differential equations (SDEs) is computationally challenging. A new forward propagation algorithm has been recently proposed for the online optimization of SDEs. The algorithm solves an SDE, derived using forward differentiation, which provides a stochastic estimate for the gradient. The algorithm continuously updates the SDE model's parameters and the gradient estimate simultaneously. This paper studies the convergence of the forward propagation algorithm for nonlinear dissipative SDEs. We leverage the ergodicity of this class of nonlinear SDEs to characterize the convergence rate of the transition semi-group and its derivatives. Then, we prove bounds on the solution of a Poisson partial differential equation (PDE) for the expected time integral of the algorithm's stochastic fluctuations around the direction of steepest descent. We then re-write the algorithm using the PDE solution, which allows us to characterize the parameter evolution around the direction of steepest descent. Our main result is a convergence theorem for the forward propagation algorithm for nonlinear dissipative SDEs.

math.PR↗

arXiv2022

Scaling Limit of Neural Networks with the Xavier Initialization and Convergence to a Global Minimum

We analyze single-layer neural networks with the Xavier initialization in the asymptotic regime of large numbers of hidden units and large numbers of stochastic gradient descent training steps. The evolution of the neural network during training can be viewed as a stochastic system and, using techniques from stochastic analysis, we prove the neural network converges in distribution to a random ODE with a Gaussian distribution. The limit is completely different than in the typical mean-field results for neural networks due to the $\frac{1}{\sqrt{N}}$ normalization factor in the Xavier initialization (versus the $\frac{1}{N}$ factor in the typical mean-field framework). Although the pre-limit problem of optimizing a neural network is non-convex (and therefore the neural network may converge to a local minimum), the limit equation minimizes a (quadratic) convex objective function and therefore converges to a global minimum. Furthermore, under reasonable assumptions, the matrix in the limiting quadratic objective function is positive definite and thus the neural network (in the limit) will converge to a global minimum with zero loss on the training set.

math.PR↗

arXiv2022

Online Adjoint Methods for Optimization of PDEs

We present and mathematically analyze an online adjoint algorithm for the optimization of partial differential equations (PDEs). Traditional adjoint algorithms would typically solve a new adjoint PDE at each optimization iteration, which can be computationally costly. In contrast, an online adjoint algorithm updates the design variables in continuous-time and thus constantly makes progress towards minimizing the objective function. The online adjoint algorithm we consider is similar in spirit to the the pseudo-time-stepping, one-shot method which has been previously proposed. Motivated by the application of such methods to engineering problems, we mathematically study the convergence of the online adjoint algorithm. The online adjoint algorithm relies upon a time-relaxed adjoint PDE which provides an estimate of the direction of steepest descent. The algorithm updates this estimate continuously in time, and it asymptotically converges to the exact direction of steepest descent as $t \rightarrow \infty$. We rigorously prove that the online adjoint algorithm converges to a critical point of the objective function for optimizing the PDE. Under appropriate technical conditions, we also prove a convergence rate for the algorithm. A crucial step in the convergence proof is a multi-scale analysis of the coupled system for the forward PDE, adjoint PDE, and the gradient descent ODE for the design variables.

math.OC↗

arXiv2021

Embedded training of neural-network sub-grid-scale turbulence models

The weights of a deep neural network model are optimized in conjunction with the governing flow equations to provide a model for sub-grid-scale stresses in a temporally developing plane turbulent jet at Reynolds number $Re_0=6\,000$. The objective function for training is first based on the instantaneous filtered velocity fields from a corresponding direct numerical simulation, and the training is by a stochastic gradient descent method, which uses the adjoint Navier--Stokes equations to provide the end-to-end sensitivities of the model weights to the velocity fields. In-sample and out-of-sample testing on multiple dual-jet configurations show that its required mesh density in each coordinate direction for prediction of mean flow, Reynolds stresses, and spectra is half that needed by the dynamic Smagorinsky model for comparable accuracy. The same neural-network model trained directly to match filtered sub-grid-scale stresses -- without the constraint of being embedded within the flow equations during the training -- fails to provide a qualitatively correct prediction. The coupled formulation is generalized to train based only on mean-flow and Reynolds stresses, which are more readily available in experiments. The mean-flow training provides a robust model, which is important, though a somewhat less accurate prediction for the same coarse meshes, as might be anticipated due to the reduced information available for training in this case. The anticipated advantage of the formulation is that the inclusion of resolved physics in the training increases its capacity to extrapolate. This is assessed for the case of passive scalar transport, for which it outperforms established models due to improved mixing predictions.

physics.flu-dyn↗

arXiv2021

Mean Field Analysis of Deep Neural Networks

We analyze multi-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. We rigorously establish the limiting behavior of the multi-layer neural network output. The limit procedure is valid for any number of hidden layers and it naturally also describes the limiting behavior of the training loss. The ideas that we explore are to (a) take the limits of each hidden layer sequentially and (b) characterize the evolution of parameters in terms of their initialization. The limit satisfies a system of deterministic integro-differential equations. The proof uses methods from weak convergence and stochastic analysis. We show that, under suitable assumptions on the activation functions and the behavior for large times, the limit neural network recovers a global minimum (with zero loss for the objective function).

math.PR↗

arXiv2021

Asymptotics of Reinforcement Learning with Neural Networks

We prove that a single-layer neural network trained with the Q-learning algorithm converges in distribution to a random ordinary differential equation as the size of the model and the number of training steps become large. Analysis of the limit differential equation shows that it has a unique stationary solution which is the solution of the Bellman equation, thus giving the optimal control for the problem. In addition, we study the convergence of the limit differential equation to the stationary solution. As a by-product of our analysis, we obtain the limiting behavior of single-layer neural networks when trained on i.i.d. data with stochastic gradient descent under the widely-used Xavier initialization.

cs.LG↗

arXiv2019

DPM: A deep learning PDE augmentation method (with application to large-eddy simulation)

Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is embedded in a partial differential equation (PDE) that expresses the known physics and learns to describe the corresponding unknown or unrepresented physics from the data. Crafted as such, the neural network can also provide corrections for erroneously represented physics, such as discretization errors associated with the PDE's numerical solution. Once trained, the deep learning PDE model (DPM) can make out-of-sample predictions for new physical parameters, geometries, and boundary conditions. Our approach optimizes over the functional form of the PDE. Estimating the embedded neural network requires optimizing over the entire PDE, which itself is a function of the neural network. Adjoint partial differential equations are used to efficiently calculate the high-dimensional gradient of the objective function with respect to the neural network parameters. A stochastic adjoint method (SAM), similar in spirit to stochastic gradient descent, further accelerates training. The approach is demonstrated and evaluated for turbulence predictions using large-eddy simulation (LES), a filtered version of the Navier--Stokes equation containing unclosed sub-filter-scale terms. The DPM outperforms the widely-used constant-coefficient and dynamic Smagorinsky models, even for filter sizes so large that these established models become qualitatively incorrect. It also significantly outperforms a priori trained models, which do not account for the full PDE. A relaxation of the discrete enforcement of the divergence-free constraint is also considered, instead allowing the DPM to approximately enforce incompressibility physics.

cs.LG↗

arXiv2019

Mean Field Analysis of Neural Networks: A Law of Large Numbers

Machine learning, and in particular neural network models, have revolutionized fields such as image, text, and speech recognition. Today, many important real-world applications in these areas are driven by neural networks. There are also growing applications in engineering, robotics, medicine, and finance. Despite their immense success in practice, there is limited mathematical understanding of neural networks. This paper illustrates how neural networks can be studied via stochastic analysis, and develops approaches for addressing some of the technical challenges which arise. We analyze one-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. We rigorously prove that the empirical distribution of the neural network parameters converges to the solution of a nonlinear partial differential equation. This result can be considered a law of large numbers for neural networks. In addition, a consequence of our analysis is that the trained parameters of the neural network asymptotically become independent, a property which is commonly called "propagation of chaos".

math.PR↗

arXiv2019

Stochastic Gradient Descent in Continuous Time: A Central Limit Theorem

Stochastic gradient descent in continuous time (SGDCT) provides a computationally efficient method for the statistical learning of continuous-time models, which are widely used in science, engineering, and finance. The SGDCT algorithm follows a (noisy) descent direction along a continuous stream of data. The parameter updates occur in continuous time and satisfy a stochastic differential equation. This paper analyzes the asymptotic convergence rate of the SGDCT algorithm by proving a central limit theorem (CLT) for strongly convex objective functions and, under slightly stronger conditions, for non-convex objective functions as well. An $L^{p}$ convergence rate is also proven for the algorithm in the strongly convex case. The mathematical analysis lies at the intersection of stochastic analysis and statistical learning.

math.PR↗

arXiv2019

Mean Field Analysis of Neural Networks: A Central Limit Theorem

We rigorously prove a central limit theorem for neural network models with a single hidden layer. The central limit theorem is proven in the asymptotic regime of simultaneously (A) large numbers of hidden units and (B) large numbers of stochastic gradient descent training iterations. Our result describes the neural network's fluctuations around its mean-field limit. The fluctuations have a Gaussian distribution and satisfy a stochastic partial differential equation. The proof relies upon weak convergence methods from stochastic analysis. In particular, we prove relative compactness for the sequence of processes and uniqueness of the limiting process in a suitable Sobolev space.

math.PR↗

arXiv2018

DGM: A deep learning algorithm for solving partial differential equations

High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, which is key since meshes become infeasible in higher dimensions. Instead of forming a mesh, the neural network is trained on batches of randomly sampled time and space points. The algorithm is tested on a class of high-dimensional free boundary PDEs, which we are able to accurately solve in up to $200$ dimensions. The algorithm is also tested on a high-dimensional Hamilton-Jacobi-Bellman PDE and Burgers' equation. The deep learning algorithm approximates the general solution to the Burgers' equation for a continuum of different boundary conditions and physical conditions (which can be viewed as a high-dimensional space). We call the algorithm a "Deep Galerkin Method (DGM)" since it is similar in spirit to Galerkin methods, with the solution approximated by a neural network instead of a linear combination of basis functions. In addition, we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic PDEs.

q-fin.MF↗

arXiv2018

Universal features of price formation in financial markets: perspectives from Deep Learning

Using a large-scale Deep Learning approach applied to a high-frequency database containing billions of electronic market quotes and transactions for US equities, we uncover nonparametric evidence for the existence of a universal and stationary price formation mechanism relating the dynamics of supply and demand for a stock, as revealed through the order book, to subsequent variations in its market price. We assess the model by testing its out-of-sample predictions for the direction of price moves given the history of price and order flow, across a wide range of stocks and time periods. The universal price formation model is shown to exhibit a remarkably stable out-of-sample prediction accuracy across time, for a wide range of stocks from different sectors. Interestingly, these results also hold for stocks which are not part of the training sample, showing that the relations captured by the model are universal and not asset-specific. The universal model --- trained on data from all stocks --- outperforms, in terms of out-of-sample prediction accuracy, asset-specific linear and nonlinear models trained on time series of any given stock, showing that the universal nature of price formation weighs in favour of pooling together financial data from various stocks, rather than designing asset- or sector-specific models as commonly done. Standard data normalizations based on volatility, price level or average spread, or partitioning the training data into sectors or categories such as large/small tick stocks, do not improve training results. On the other hand, inclusion of price and order flow history over many past observations is shown to improve forecasting performance, showing evidence of path-dependence in price dynamics.

q-fin.ST↗

arXiv2018

Deep Learning for Mortgage Risk

We develop a deep learning model of multi-period mortgage risk and use it to analyze an unprecedented dataset of origination and monthly performance records for over 120 million mortgages originated across the US between 1995 and 2014. Our estimators of term structures of conditional probabilities of prepayment, foreclosure and various states of delinquency incorporate the dynamics of a large number of loan-specific as well as macroeconomic variables down to the zip-code level. The estimators uncover the highly nonlinear nature of the relationship between the variables and borrower behavior, especially prepayment. They also highlight the effects of local economic conditions on borrower behavior. State unemployment has the greatest explanatory power among all variables, offering strong evidence of the tight connection between housing finance markets and the macroeconomy. The sensitivity of a borrower to changes in unemployment strongly depends upon current unemployment. It also significantly varies across the entire borrower population, which highlights the interaction of unemployment and many other variables. These findings have important implications for mortgage-backed security investors, rating agencies, and housing finance policymakers.

q-fin.ST↗