Probabilistic Inverse Modeling An Application in Hydrology Somya SharmaRahul GhoshArvind RenganathanXiang Li Snigdhansu ChatterjeeJohn NieberChristopher DuyVipin Kumar

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Probabilistic Inverse Modeling: An Application in Hydrology

Somya Sharma∗Rahul Ghosh∗Arvind Renganathan∗Xiang Li∗

Snigdhansu Chatterjee∗John Nieber∗Christopher Duﬀy+Vipin Kumar∗

Abstract

Rapid advancement in inverse modeling methods have

brought into light their susceptibility to imperfect data. The

astounding success of these methods has made it impera-

tive to obtain more explainable and trustworthy estimates

from these models. In hydrology, basin characteristics can

be noisy or missing, impacting streamﬂow prediction. For

solving inverse problems in such applications, ensuring ex-

plainability is pivotal for tackling issues relating to data

bias and large search space. We propose a probabilistic in-

verse model framework that can reconstruct robust hydrol-

ogy basin characteristics from dynamic input weather driver

and streamﬂow response data. We address two aspects of

building more explainable inverse models, uncertainty esti-

mation and robustness. This can help improve the trust of

water managers, handling of noisy data and reduce costs.

We propose uncertainty based learning method that oﬀers

6% improvement in R2for streamﬂow prediction (forward

modeling) from inverse model inferred basin characteristic

estimates, 17% reduction in uncertainty (40% in presence of

noise) and 4% higher coverage rate for basin characteristics.

1 Introduction

Researchers in scientiﬁc communities study engineered

or natural systems and their responses to external

drivers. In hydrology, streamﬂow prediction is one cru-

cial research problem for understanding hydrology cy-

cles, ﬂood mapping, water supply management, and

other operational decisions. For a given entity (river-

basin/catchment), the response (streamﬂow) is gov-

erned by external drivers (meteorological data) and

complex physical processes speciﬁc to each entity (basin

characteristics). Process-based models are commonly

used to study streamﬂow in river basins (for example,

Soil & Water Assessment Tool). However, these hy-

drological models are constrained by assumptions, con-

tain many parameters that need calibration and incur

enormous computation cost. In addition, these mod-

els are often calibrated on every speciﬁc catchment and

thus can require speciﬁc ﬁne-tuning for each basin. As

promising alternatives, machine learning (ML) models

are increasingly being used [30] (Figure 1 shows the di-

∗University of Minnesota - Twin Cities. {sharm636, ghosh128,

renga016, lixx5000, chatt019, nieber, kumar001}@umn.edu, +

Pennsylvania State University {cxd11}@psu.edu

Forward Model

Figure 1: Forward model using weather drivers xt

iand

river basin characteristics zt

ito estimate streamﬂow yt

[16]

agrammatic representation of this data-driven forward

model). In our study, an entity’s response to external

drivers depends on its inherent properties (called entity

characteristics). For example, for the same amount of

precipitation (external driver), two river basins (enti-

ties) will have very diﬀerent streamﬂow (response) val-

ues depending on their land-cover type (entity char-

acteristic) [38]. Disregarding these inherent proper-

ties of entities can lead to sub-optimal model per-

formance. Knowledge-guided self-supervised learning

(KGSSL) [16] has been proposed to extract these en-

tity characteristics using the input drivers and output-

response data.

Developing such entity-aware inverse models re-

quires addressing several challenges. Often, the mea-

sured characteristics are only surrogate variables for the

actual entity characteristics, leading to inconsistencies

and high uncertainty. Uncertainty can arise due to sev-

eral reasons, such as measurement error, missing data,

and temporal changes in characteristics. Moreover, in

real-world applications these characteristics may be es-

sential in modeling the driver-response relation. How-

ever, they may be completely unknown, not well un-

derstood, or not present in the available set of entity

characteristics. A principled method of managing this

uncertainty due to imperfect data can contribute in im-

proving trust of data-driven decision making from these

methods.

In this paper, we introduce uncertainty quantiﬁca-

tion in learning representations of static characteristics.

Such a framework can help quantify the eﬀect of multi-

ple sources of uncertainty that introduce bias and er-

ror in decision-making. For instance, Equiﬁnality of

hydrological modeling (diﬀerent model representation

result in same model results) is a widely known phe-

nomenon aﬀecting the adoption of hydrology models in

Unauthorized reproduction of this article is prohibited

arXiv:2210.06213v1 [cs.LG] 12 Oct 2022

Figure 2: KGSSL for representation learning. The forward model learns streamﬂow (y) as functional

approximation of weather drivers (x) and river basin static attributes (z). KGSSL leverages an inverse modeling

framework for learning robust static attribute estimates (ˆz). The robust estimates aid in improving prediction

performance of the forward model.

practice [20]. Uncertainty in model structure and input

data are also widespread. In real world applications,

studying these can help improve trust of water man-

agers, improve hydrological process understanding, re-

duce costs and make predictions more explainable and

robust [36]. To achieve this, we propose a probabilis-

tic inverse model for simultaneously learning represen-

tations of static characteristics and quantifying uncer-

tainty in these predictions. As a consequence, we ana-

lyze the framework’s reconstruction capabilities and its

susceptibility to adversarial perturbations - our model

is able to maintain same level of robustness as KGSSL.

We also propose an uncertainty based learning (UBL)

method to reduce epistemic uncertainty (uncertainty in

predictions due to imperfect model and imperfect data)

in our reconstructions. We show that it results in reduc-

ing the temporal artifacts in static characteristic pre-

dictions by 17% and also reduces the epistemic uncer-

tainty by 36%. We also demonstrate the improvement

in streamﬂow prediction (in the forward model) using

these reconstructed static characteristics (6% increase

in test R2). We provide model performance for recon-

struction and forward modeling and compare it against

the baselines, KGSSL [16] and CT-LSTM [30], state-

of-the-art for streamﬂow prediction. Since, we use a

probabilistic model for estimating static characteristics,

we obtain a posterior distribution instead of point es-

timates. This enables us to compute coverage rate of

how often the observed values lie within the bounds of

the inferred static characteristics’ posterior prediction

distribution. In practice, this can help water managers

and public understand if we can reliably obtain a close

enough prediction, even if we are not always accurate

- analysis that can not be done with the deterministic

inverse model. UBL oﬀers a 4% increase in coverage

rate.

2 Related Work

Robustness: In several large-scale applications in areas

like computer vision and natural language processing,

presence of even small, imperceptible adversarial per-

turbation can exacerbate model performance [10,48,51].

While, several of the robustness studies focus on the ef-

fect of diﬀerent noises [32, 57], many other studies fo-

cus on methods to mitigate the adverse eﬀects of these

perturbations [35, 42, 50]. Through objective modiﬁca-

tion [11,24,26] or propagation of input-output relation-

ship constraints [9,21], deep learning architectures have

been modiﬁed to improve robustness. In real-world data

sets, natural variations (like blurring) have been stud-

ied [41, 47]. Issues like adversarial transferability [22]

and data brittleness issues (robustness issues due to

overﬁtting [40]) bring to light the limitations of mod-

ern machine learning methods. More recent studies also

look at Bayesian deep learning models for their robust-

ness properties [4, 5, 43].

Inverse Problems: In physical sciences [7,27,39,55],

several recent advances have focused on solving inverse

problems. Unlike standard inversion methods in math-

ematics, that rely on non-linear optimization for cal-

culating inverse of a forward model, recent machine

learning methods allow us to learn the inverse mapping

from datasets. This makes it imperative to mitigate

any representation error and data biases before solving

the inverse problem [2]. Further, within these vast ar-

ray of methodologies, selection of the right method is

crucial - since, searching for an inverse mapping may

be diﬃcult due to the large search space. Bayesian op-

timization for searching and iterative gradient descent

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ProbabilisticInverseModeling:AnApplicationinHydrologySomyaSharma*RahulGhoshArvindRenganathanXiangLiSnigdhansuChatterjeeJohnNieberChristopherDuy+VipinKumarAbstractRapidadvancementininversemodelingmethodshavebroughtintolighttheirsusceptibilitytoimperfectdata.Theastoundingsuccessofthesemethodsha...

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Probabilistic Inverse Modeling An Application in Hydrology Somya SharmaRahul GhoshArvind RenganathanXiang Li Snigdhansu ChatterjeeJohn NieberChristopher DuyVipin Kumar

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