PHYSICS -INFORMED NEURAL NETWORKS OF THE SAINT -VENANT EQUATIONS FOR DOWNSCALING A LARGE -SCALE RIVER MODEL

2025-05-02 0 0 5.54MB 28 页 10玖币

侵权投诉

PHYSICS-INFORMED NEURAL NETWORKS OF THE

SAINT-VENANT EQUATIONS FOR DOWNSCALING A

LARGE-SCALE RIVER MODEL

Dongyu Feng

Paciﬁc Northwest National Laboratory

Richland, WA 99354

dongyu.feng@pnnl.gov

Zeli Tan

Paciﬁc Northwest National Laboratory

Richland, WA 99354

zeli.tan@pnnl.gov

QiZhi He

University of Minnesota

Minneapolis, MN 55455

qzhe@umn.edu

ABSTRACT

Large-scale river models are being reﬁned over coastal regions to improve the scientiﬁc understanding

of coastal processes, hazards and responses to climate change. However, coarse mesh resolutions and

approximations in physical representations of tidal rivers limit the performance of such models at

resolving the complex ﬂow dynamics especially near the river-ocean interface, resulting in inaccurate

simulations of ﬂood inundation. In this research, we propose a machine learning (ML) framework

based on the state-of-the-art physics-informed neural network (PINN) to simulate the downscaled

ﬂow at the subgrid scale. First, we demonstrate that PINN is able to assimilate observations of various

types and solve the one-dimensional (1-D) Saint-Venant equations (SVE) directly. We perform

the ﬂow simulations over a ﬂoodplain and along an open channel in several synthetic case studies.

The PINN performance is evaluated against analytical solutions and numerical models. Our results

indicate that the PINN solutions of water depth have satisfactory accuracy with limited observations

assimilated. In the case of ﬂood wave propagation induced by storm surge and tide, a new neural

network architecture is proposed based on Fourier feature embeddings that seamlessly encodes the

periodic tidal boundary condition in the PINN’s formulation. Furthermore, we show that the PINN-

based downscaling can produce more reasonable subgrid solutions of the along-channel water depth

by assimilating observational data. The PINN solution outperforms the simple linear interpolation

in resolving the topography and dynamic ﬂow regimes at the subgrid scale. This study provides

a promising path towards improving emulation capabilities in large-scale models to characterize

ﬁne-scale coastal processes.

1 Introduction

The population growth near coastal regions increases human exposure to natural hazards of hurricanes and ﬂooding [

Coastal inundation area changes greatly during extreme ﬂooding events. The ﬂow and ﬂuxes at the terrestrial-aquatic

interface or tidal rivers are governed by river discharge, tide and storm surge, as well as their complex physical

interactions [

], which could also impact sediment dynamics and biogeochemical processes [

]. For example, during

landfalling with tropical cyclones bringing heavy precipitation, the ﬂood wave propagation from downstream storm

surge and tide impedes the upstream discharge and this backwater effect in turn modulates the water level [

]. Such

ﬂood event is referred as compound ﬂooding as multiple mechanisms can occur simultaneously or in close successions

[

]. Climate warming will further exacerbate risks of such hazards because it will create more dynamic ﬂooding

zones in river systems over the low-lying regions [

] by intensifying extreme storm surge and precipitation [

accelerating sea level rise (SLR) [

] and enhancing tidal dynamics [

]. It thus becomes crucial to characterize the

complex interactions between these processes in a warming climate.

The mitigation of compound ﬂood risks requires the understanding of tidal river dynamics, which are affected by the

nonlinear interaction of hydrological and hydrodynamical processes, river geometry and local basin characteristics [

Beyond the traditional harmonic tidal analysis, there is a growing application of using numerical models to study the

compounding dynamics. For example, hydraulic or hydrodynamic models are conﬁgured in tidal estuaries to simulate

arXiv:2210.03240v1 [physics.flu-dyn] 6 Oct 2022

coastal inundation and ﬂood wave propagation in river networks at the event scale [

] or under a changing

climate [

]. The upstream and coastal boundary conditions of these models are provided by hydrological and coastal

models, respectively. There are also studies that apply more efﬁcient river hydrological models [

], coupled with

global tide and surge models [

], in larger-scale risk assessments [

]. However, these modeling studies usually

have not fully utilized gauged measurements that are abundant in tidal rivers and high-resolution remote sensing data.

Such data, if appropriately assimilated, will likely improve the model performance, especially for the large-scale models

that under-resolve the river dynamics.

Large-scale river models are a major component of Earth System Models (ESMs) that are the primary tool for climate

change research. Such models have been used extensively in global water cycle simulations [

] and large-scale ﬂood

risk assessments [

]. Although local river models are frequently used to assess the compound ﬂooding induced by

pluvial, ﬂuvial and coastal processes [

], the large-scale models are preferred at regional to global scales [

However, the approximations in the governing physics and the deﬁciency in mesh resolutions limit these models in

representing the local ﬂooding processes, particularly near the highly dynamic terrestrial-aquatic interface. While

the large-scale river models can conserve water mass and simulate reasonable river discharge at the global scale [

these reduced-physics models employ simpliﬁed forms for momentum propagation, which requires less data in river

channel topography and lower computational cost than dynamic models [

]. These approximations inevitably limit the

representation of physical characteristics of dynamic systems [10].

Speciﬁcally, in a one-dimensional (1-D) river system, the ﬂow dynamics is governed by the 1-D Saint-Venant equations

(SVE). Solving SVE in dynamic river models is computationally demanding, because explicit numerical schemes

usually require shorter time steps to converge and implicit schemes need to evaluate SVE equations multiple iterations

at every time step [

]. While it appears feasible to apply SVE to a large scale at high spatial resolutions with the

advancement of high performance computing (HPC), such dynamic models require high-quality river geometry ﬁelds.

This requirement cannot be met by the large-scale river models [

]. As a result, approximations must be made to SVE

to relax model requirements. For example, for the two common SVE approximations, the diffusive wave equation

neglects the Eulerian inertial acceleration and assumes that gravity, friction and pressure dominate the momentum,

while the local inertial equation [

] only neglects the advection term. These assumptions, however, are likely not valid

for ﬂood-prone river systems. In a ﬂat sloping or tidally-inﬂuenced river system, ﬂood wave dominates momentum

propagation as velocity changes rapidly with space and time. The inertial approximation reduces the ﬂood propagation

speed, attenuates the water surface gradient [

] and underestimates the physical characteristics such as the raising and

recession of limbs [35].

Even with simpliﬁed physics, large-scale river models have to apply low-resolution (5 km

∼

25 km) meshes to

accommodate the high computational cost in large-scale applications. For example, the river component of Energy

Exascale Earth System Model (E3SM) applies a resolution of 12.5 km [

], which is too coarse to resolve the ﬂow

dynamics induced by the change of ﬂow regimes over various topographical features. Additionally, in such models,

ﬂood inundation over a ﬂoodplain is usually simulated using mass-conserved schemes that assume no between-cell

exchange of inundated water [

]. Such schemes, despite achieving satisﬁed estimation of ﬂood fraction within

a cell, cannot represent within-cell heterogeneity of water depth, because the latter would never be known without

resolving subgrid ﬂow dynamics.

While extensive efforts have been made to develop efﬁcient numerical schemes to represent a more dynamic river

system in large-scale models [

], the alternative is to derive downscaled solutions at the subgrid scale in regions

of interest using statistical or machine learning (ML) approaches. Downscaling coarse-scale numerical models or

forcing data to a ﬁne-scale solution is an established research area in climate and ocean modeling [

]. The

downscaling is often classiﬁed into two categories: statistical downscaling and dynamic downscaling. The statistical

downscaling usually ﬁrst builds a statistical relationship between the coarse-scale model and local data and then uses

the relationship to produce high spatial- or temporal-resolution outputs [

]. In the dynamic downscaling, a local

high-resolution model simulation is performed with inputs extrapolated from large-scale processes (e.g., boundary

conditions from a large-scale model) to produce high-resolution outputs directly [

]. The application of downscaling

to river modeling is still limited. Most previous studies only focused on downscaling a single or grouped gauged

measurements to a higher temporal resolution statistically for assessing the local climate change impacts [

], or

downscaling two-dimensional inundation maps by overlaying water depth computed from a coarse-resolution model or

satellite data onto a high-resolution digital elevation model (DEM) [

]. To our knowledge, the only method to

dynamically downscale the 1-D channel ﬂow is to linearly interpolate the modeled water depth at each grid node and

available gauged observations across in-between cross-sections [

]. This method does not require channel topology

information and cannot resolve the spatially-varied ﬂow regimes within a grid cell. To ﬁll this gap, this work develops a

ML-based downscaling method that efﬁciently assimilates available observations and computes subgrid solutions.

Over recent years, the availability of water depth data increases rapidly as such data are consistently collected from

in-situ measurements at gauging stations as well as by remote sensing [

], such as satellite radar altimetry [

]. In-situ

data are continuous in time and sparse in space [

]. In contrast, remote sensing data, while measured at large time

intervals, can cover great spatial domains at high resolutions [

]. For example, the upcoming Surface Water and

Ocean Topography (SWOT) will enable direct measurements of water depth at a spatial resolution of 50 m for all rivers

wider than 100 m [

]. Such data could potentially be directly incorporated into ML models to obtain downscaled

solutions [

]. Remote sensing and gauged observation data, along with numerical solutions, provide the basis for the

development of ML-based methods.

The ML technique has been explored to alleviate the limitations in the river modeling from various aspects. For example,

ML models are employed as surrogates or emulators of hydrological models to perform rapid simulations for real-time

predictions and model calibrations [

]. ML models are also proposed to predict hydrograph [

], maximum water

levels [

], inﬁltration [

], as well as real-time inundation maps [

]. Despite differences in implementation, these

methods, such as regression trees, random forests and neural networks, all treat the ML model as a black box. Model

predictions are made based on a set of input variables, such as precipitation and streamﬂow. Additionally, training an

emulator is usually expensive as it requires abundant including observations or multiple realizations of high-resolution

numerical simulations. As a result, some alternative ML algorithms are proposed to replace computationally demanding

components of a numerical scheme with a computationally efﬁcient ML model. For example, the convolutional neural

network (CNN) is used to solve the Euler equation for iterative pressure correction in the velocity update algorithm

[

]. Similarly, artiﬁcial neural networks are used to estimate the velocity in the inertial wave equation [

]. However,

these ML models are still computationally demanding for training and also they require accurate training data to avoid

bad numerical solutions.

Emerged recently are the state-of-the-art physics-informed ML models. Such models are trained by enforcing the

relevant physics. Data, mathematical models and domain knowledge are seamlessly integrated and implemented through

regression networks [

]. Physics-informed neural networks (PINN) solve partial differential equations (PDEs) using

feed-forward neural network architectures and respect the corresponding physical laws [

]. The unknown solutions of

governing equations are inferred from the initial and boundary data of the state variables. The ML models trained with

physical constraints require much less data than traditional ML emulators. This approach thus greatly alleviates the

limitations of ML in model training caused by data scarcity. The PINN algorithm can also be used in inverse problems to

estimate parameters of a PDE with observations of state variables provided [

], as well as in developing surrogate

models from sparse data [

]. Specialized network architectures have been developed for PINN to meet speciﬁc

physical laws [

], accelerate the training process [

] and improve the model accuracy [

]. As the result, many new

variants of PINN were introduced recently, such as multiphysics PINN [

], Parareal PINN [

], and DeepONet

[75]. For a comprehensive review of PINN, please refer to [63].

Due to the illustrated strengths, PINN has been explored extensively in solving forward, inverse and data assimilation

problems in science and engineering [

], such as subsurface transport in porous media [

] , acoustic wave propagation

[

], solid mechanics [

] and heat transfer [

]. In particular, PINN demonstrates tremendous success in simulating

ﬂow dynamics [

], including high-speed aerodynamic ﬂows [

], incompressible ﬂow governed by Navier Stokes

equations [

], laminar ﬂows at low Reynolds numbers [

], and even ﬂuid-structure interaction [

]. Previous

applications of PINN to the hydraulic modeling [

], despite limited, show that it can outperform Artiﬁcial Neural

Network (ANN) in ﬂood simulations [85].

The ML-based downscaling is essentially a data assimilation problem with the ML models merging measurements and

numerical model outputs into the ML training. A key feature of PINN is its simplicity for assimilating observations

[

]. The time-varying observations and/or spatial snapshots at various spatiotemporal scales can be readily

incorporated into the PINN training. Based on this feature, we propose a data assimilation approach based on the

standard PINN framework and its variant to address problems in Fourier spaces (ff-PINN) [

]. The proposed PINN

computes the subgrid solution of a coarse model output and assimilated observations without modifying the numerical

algorithms or reﬁning the mesh resolution. By incorporating the dynamics presented in the measurements, PINN is able

to emulate the nonlinear interactions of storm surge, tide and discharge within tidal rivers. We do not intend to replace

the numerical solver by PINN. Instead, we aim to build the PINN method upon the infrastructure of existing large-scale

models to provide downscaled solutions at cells of interest.

The PINN-based method has a few additional advantages over local-scale numerical models. PINN is mesh-free,

making it an efﬁcient and ﬂexible tool for downscaling. In practice, the downscaled solution at regions of interest may

be obtained by conﬁguring a local numerical model or an emulator at ﬁne scales. Even though PINN itself may not be

as efﬁcient as the local model in terms of running speed, PINN does not require discretization and thus saves efforts

of generating meshes for the numerical conﬁguration. The meshless formalism enables ﬂexible implementation in

various domains as the partial derivatives are evaluated using automatic differentiation in Tensorﬂow [

]. Moreover,

the PINN-based data assimilation allows exploring the potentials of merging various types of measurements from

hydroacoustic meters, remote sensing platforms and other data collection instruments.

This research aims at developing a downscaling approach based on the PINN data assimilation to resolve subgrid

variability of river dynamics in coastal regions for large-scale river models. A PINN solver of 1-D SVE is developed

to merge in-situ and remote sensing measurements as well as coarse-scale model solutions to obtain a more accurate

downscaled solution near the coastal interface and improve ﬂood simulation. To the best of our knowledge, this is

the ﬁrst study in literature which develops a data assimilation model for downscaled river ﬂow characterized by SVE

with the utilization of physics-informed machine learning methods. Several synthetic cases are tested to demonstrate

our model’s capability of reproducing ﬂood propagation in a theoretical setup. Importantly, we develop a new neural

network architecture to account for periodic tidal oscillations at the river downstream boundary and investigate the

effects of the proposed network architecture on the PINN performance. This manuscript is organized as follows. In

Section 2, we present the PINN method for solving SVE and data assimilation. The PINN-based downscaling is

also discussed. The problem formulation of each case study and the corresponding results are provided in Section 3.

Section 4 presents the result discussions, potential limitations in realistic applications and the future path. Finally, the

conclusions are provided in Section 5.

2 Methodology

2.1 Saint-Venant Equations

The 1-D SVE consists of a continuity equation and a momentum equation for the dynamics of velocity (

) and water

depth (h) along the river channel. The incompressible continuity equation is deﬁned as

∂h

∂t +u∂h

∂x =q, (1)

where

is distance along the river channel,

is time and

is the water inﬂow per unit length of the channel from land

surface and subsurface runoff, groundwater and precipitation. Throughout the study, it is assumed that the river channel

does not receive water from the land and atmosphere (q= 0). The momentum equation in the full dynamic form is

∂u

∂t +u∂u

∂x +g∂h

∂x +g(Sf−S)=0,(2)

where

is gravity,

is riverbed slope and

is the friction slope that can calculated using the Chezy–Manning equation:

Sf=n2|u|u

,(3)

in which Manning’s roughness coefﬁcient

is used as the frictional coefﬁcient and

is the hydraulic radius. This

study only considers rectangular channels, so Rcan be expressed as

R=bh/(2h+b),(4)

where

is the channel width. The large-scale river models usually use simpliﬁed forms of the momentum equation,

including the local inertial equation that neglects the convective acceleration term (

u∂u

∂x

) [

], the diffusive wave

equation that neglects both the convective acceleration term and the local acceleration term (

∂u

∂t

) [

], and the kinematic

wave equation that neglects all partial derivative terms [

]. None of these simpliﬁed schemes include advection that

dominates momentum propagation when the ﬂood wave dynamics is strong [5].

2.2 PINN approximation of SVE

In this study, instead of using numerical discretization, we take the ﬁrst attempt to solve SVE using the PINN method.

As illustrated in the schematic diagram (Figure 1(a)), the fully connected feed-forward deep neural network (DNN) in

PINN takes spatial and temporal coordinates

and

as inputs and predicts the corresponding unknown variables

and

in the output layer. There are

hidden layers between the input and output layers and

neurons in each hidden layer.

The neurons between the adjacent layers are fully connected and the inputs of the

lth

layer (

) are fed from the outputs

of the previous layer (zl−1):

zl=σ(Wlzl−1+bl),(5)

where the hyper-parameters

and

are the weight matrix and bias vector at the

lth

layer, determined after training,

and

is the activation function used to introduce nonlinearity to each output component. In this study, the activation

function is selected as

tanh(x)

, and

and

are initialized using a widely-used initialization scheme, Xavier scheme

[88], where initial weights are sampled from a truncated normal distribution.

The partial derivatives in the governing equations are used with the solutions predicted by DNN to approximate

the residuals of the governing equations. The partial derivatives are obtained using the automatic differentiation,

implemented in the deep learning platform Tensorﬂow [

] that computes the gradient of an output variable with respect

to the input coordinates [89].

In PINN, the solutions of SVE, the instantaneous velocity u(x, t)and water depth h(x, t), are estimated with:

u(x, t)≈ˆu(x, t, θ),(6)

h(x, t)≈ˆ

h(x, t, θ),(7)

where xand tare the space and time vectors, and θis the vector of weights and biases [90].

The loss function consists of the residuals of PDE (i.e. the summed residual errors of equation 1 and 2 when

ˆu

and

are substituted), and the mean square errors of the DNN approximation against the boundary condition (BC) and the

snapshot data associated with uand h(the snapshot data is denoted with symbol S):

J(θ) = Jf(θ) + X

wjJj(θ).(8)

The subscript

BCu

BCh

and

represent the loss terms and weighting coefﬁcients correspondingly.

The PDE loss (Jf) is deﬁned as

Jf(θ) = 1

i=1

|rf2(xi, ti, θ)|,(9)

where

denotes the number of collocation points in the computational domain. The residual of SVE (

) is obtained

by summing the continuity equation

(1)

and the momentum equation

(2)

with

ˆu

and

substituted. The loss functions

corresponding to BC and spatial snapshot data are deﬁned as:

JBCu(θ) = 1

NBCu

i=1

|ˆu(xi, ti, θ)−u(xi, ti)|2,

JBCh(θ) = 1

NBCh

i=1

|ˆ

h(xi, ti, θ)−h(xi, ti)|2,x∈ΩBC , t ∈[0, T ](10)

JSu(θ) = 1

NSu

i=1

|ˆu(xi,0, θ)−u(xi,0)|2,

JSh(θ) = 1

NSu

i=1

|ˆ

h(xi,0, θ)−h(xi,0)|2,x∈ΩS(11)

where

NBC

refers to the number of points sampled at the boundaries of the computational domain and

is the number

of points sampled at the spatial snapshots, and

is the simulation period. Time-varying data, such as the observations

and

, can be used in the PINN training to add further constraints. In this study, we add two additional terms (i.e.

wobsuJobsuand wobshJobsh) to the total loss in equation 8. The loss functions corresponding to the observations are

Jobsu(θ) = 1

Nobsu

i=1

|ˆu(xi, ti, θ)−u(xi, ti)|2,

Jobsh(θ) = 1

Nobsh

i=1

|ˆ

h(xi, ti, θ)−h(xi, ti)|2,x∈Ωobs, t ∈[0, T ](12)

Previous studies showed that the weighting coefﬁcients of the loss functions (

) in equation

(8)

are critical to the PINN

training as these coefﬁcients are used to balance the contribution of different loss terms [

]. Imbalanced

loss terms can lead to failure in PINN. The selection of weights is problem-speciﬁc because the optimal combination

of weights varies across different ﬂow scenarios, depending on system properties and conditions. The weights are

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

PHYSICS-INFORMEDNEURALNETWORKSOFTHESAINT-VENANTEQUATIONSFORDOWNSCALINGALARGE-SCALERIVERMODELDongyuFengPacicNorthwestNationalLaboratoryRichland,WA99354dongyu.feng@pnnl.govZeliTanPacicNorthwestNationalLaboratoryRichland,WA99354zeli.tan@pnnl.govQiZhiHeUniversityofMinnesotaMinneapolis,MN55455qzhe@umn....

展开>> 收起<<

PHYSICS -INFORMED NEURAL NETWORKS OF THE SAINT -VENANT EQUATIONS FOR DOWNSCALING A LARGE -SCALE RIVER MODEL.pdf

共28页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

PHYSICS -INFORMED NEURAL NETWORKS OF THE SAINT -VENANT EQUATIONS FOR DOWNSCALING A LARGE -SCALE RIVER MODEL

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: