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SUMMARY:Machine learning for PDEs workshop 2022
DESCRIPTION:The goal of the workshop is to gather experts from different ar
eas in this inter-disciplinary field to investigate and discuss how to har
ness the power of machine learning techniques to solve high-dimensional\,
non-linear partial differential equations (PDEs)\, as well as how to lever
age the theory of PDEs to construct better machine learning models and stu
dy their theoretical properties.\nPDEs are a dominant modelling paradigm u
biquitous throughout science\, from fluid dynamics to quantum mechanics\,
to calculus of variations and quantitative finance. When the PDEs at hand
are low dimensional (dim=1\,2\,3\,4) they can generally be solved numerica
lly leveraging a large arsenal of techniques developed over the last 150 y
ears\, including finite difference and finite elements methods.\nNonethele
ss\, many PDEs arising from complex\, real world financial-engineering-phy
sical problems are often so high-dimensional (sometimes even infinite dime
nsional) that classical numerical techniques are either not directly appli
cable or do not scale to high-resolution computations. Examples of such in
tractable equations include pricing and hedging with rough volatility pric
e dynamics\, non-Markovian\, path-dependent stochastic control problems\,
and turbulent fluid flow dynamics to be solved on very fine scales.\nRecen
t advances in Machine Learning (ML) have enabled for the development of no
vel computational techniques for tackling PDE-based problems considered un
resolvable with classical methods. Physics-informed neural networks\, neur
al differential equations\, and neural operators are among the most popula
r models used to tackle PDE-related problems with deep learning.\nThe goal
of this workshop is to develop a classification of ML techniques dependin
g on the type of PDE and to set clear new directions in the design of opti
mal numerical schemes\, both numerically and theoretically (with convergen
ce results). The list of participants is designed to maximise inter-discip
linarity and encourage diversity with experts in different fields\, such a
s stochastic analysis\, numerical analysis\, mathematical finance and mach
ine learning.\nOrganisers\nDr A. Jacquier (Imperial College)\, Prof. J. Ru
f (LSE) and Dr C. Salvi (Imperial College).\nPlease contact the organisers
if you are interested in attending the workshop.\nFunding sources\nEPSRC\
, LSE\, Imperial.\nConfirmed speakers and schedule\n \n\n\n\n\nTuesday 6t
h\n13:30-13:45\nWelcome Speech\n\n\n\n13:45-14:45\nAnastasia Borovykh\n\n\
n\n14:45-15:45\nYuri Saporito\n\n\n\n15:45-16:15\nCoffee Break\n\n\n\n16:1
5-17:15\nYufei Zhang\n\n\nWednesday 7th\n10:15-11:15\nMartin Larsson\n\n\n
\n11:15-11:45\nMarc Sabate Vidales\n\n\n\n11:45-12:30\nSam Cohen\n\n\n\n12
:30-13:45\nLunch Break\n\n\n\n13:45-14:45\nStefania Fresca\n\n\n\n14:45-15
:45\nChristoph Reisinger\n\n\n\n15:45-16:15\nCoffee Break\n\n\n\n16:15-17:
15\nDante Kalise\n\n\n\n19:00-22:00\nWorkshop Dinner (by invitation only)\
n\n\nThursday 8th\n10:00-10:30\nAthena Picarelli\n\n\n\n10:30-11:30\nCris
Salvi\n\n\n\n11:30-11:45\nCoffee Break\n\n\n\n11:45-12:45\nCamilo Garcia T
rillos\n\n\n\n \n\nTitles and abstracts\nAnastasia Borovykh\nTitle: On th
e choice of loss functions and initializations for deep learning-based sol
vers for PDEs\nAbstract: In this talk we will discuss several challenges t
hat arise when solving PDEs with deep learning-based solvers. We will begi
n with defining the loss function of a general PDE and discuss how this ch
oice of loss function\, and specifically the weighting of the different lo
ss terms\, can impact the accuracy of the solution. We will show how to ch
oose an optimal weighting that corresponds to accurate solutions. Next\, w
e will focus on the approximation of the Hamilton-Jacobi-Bellman partial d
ifferential equation associated to optimal stabilization of the NonLinear
Quadratic Regular Problem. It is not obvious that the neural network will
converge to the correct solution with just any type of initialisation\; th
is is particularly relevant when the solution to the HJB-PDE is non-unique
. We will discuss a two-step learning approach where the model is pre-trai
ned on a dataset obtained from solving a state-dependent Riccati equation
and we show that in this way efficient and accurate convergence can still
be obtained.\n\nSam Cohen\nTitle: Neural Q-learning solutions to elliptic
PDEs\nAbstract: Solving high-dimensional partial differential equations (P
DEs) is a major challenge in scientific computing. We develop a new numeri
cal method for solving elliptic-type PDEs by adapting the Q-learning algor
ithm in reinforcement learning. Using a neural tangent kernel (NTK) approa
ch\, we prove that the neural network approximator for the PDE solution\,
trained with the Q-PDE algorithm\, converges to the trajectory of an infin
ite-dimensional ordinary differential equation (ODE) as the number of hidd
en units becomes infinite. For monotone PDE (i.e. those given by monotone
operators)\, 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 increase
s. The numerical performance of the Q-PDE algorithm is studied for several
elliptic PDEs. Based on joint work with Deqing Jiang and Justin Sirignano
.\nStefania Fresca\nTitle: Deep learning-based reduced order models for sc
ientific applications\nAbstract: The solution of differential equations by
means of full order models (FOMs)\, such as\, e.g.\, the finite element m
ethod\, entails prohibitive computational costs when it comes to real-time
simulations and multi-query routines. The purpose of reduced order modeli
ng is to replace FOMs with suitable surrogates\, so-called reduced order m
odels (ROMs)\, characterized by much lower complexity but still able to ex
press\nthe physical features of the system under investigation. Convention
al ROMs anchored to the assumption of modal linear superimposition\, such
as proper orthogonal decomposition (POD)\, may reveal inefficient when dea
ling with nonlinear time-dependent parametrized PDEs\, especially for prob
lems featuring coherent structures propagating over time. To enhance ROM e
fficiency\, we propose a nonlinear approach to set ROMs by exploiting deep
learning (DL) algorithms\, such as convolutional neural networks (CNNs).
In the resulting DL-ROM\, both the nonlinear trial manifold and the nonlin
ear reduced dynamics are learned in a non-intrusive way by relying on DL a
lgorithms trained on a set of FOM snapshots\, obtained for different param
eter values. Furthermore\, in case of large-scale FOMs\, a former dimensio
nality reduction on FOM snapshots through POD enables to speed-up training
times and to substantially decrease the network complexity. Accuracy and
efficiency of the DL-ROM technique are assessed in different scientific ap
plications aiming at solving parametrized PDE problems\, e.g.\, in cardiac
electrophysiology\, computational mechanics and fluid dynamics\, possibly
accounting for fluid-structure interaction effects\, where new queries to
the DL-ROM can be computed in real-time. Finally\, with the aim of moving
towards a rigorous justification on DL-ROMs mathematical foundations\, er
ror bounds are derived for the approximation of nonlinear operators by mea
ns of CNNs. The resulting error estimates provide a clear interpretation o
f the hyperparameters defining the neural network architecture.\nCamilo Ga
rcia Trillos\nTitle: Neural network approximation of some second order sem
ilinear PDEs\nAbstract: Since its inception in the early 90s\, the well-kn
own connection between second-order semi-linear PDEs and Markovian BSDEs h
as been useful in creating numerical probabilistic methods to solve the fo
rmer. Our approach to the solution of these PDEs belongs to a recent strea
m in the literature that uses neural networks together with the BSDE conne
ction to define numerical methods that are robust and efficient in large d
imensions. In contrast with existing works\, our analysis focuses on the c
ase where derivatives enter ‘quadratically’ in the semilinear term\, c
overing some interesting cases in control theory. In this setting\, we stu
dy both forward and backward-types of neural network based methods. Joint
work with Daniel Bussell.\nDante Kalise\nTitle: Data-driven schemes for Ha
milton-Jacobi-Bellman equations\nAbstract: In this talk I will discuss dif
ferent computational aspects arising in the construction of data-driven sc
hemes for HJB PDEs. First\, I will discuss synthetic data generation throu
gh representation formulas including Pontryagin’s Maximum Principle and
State-dependent Riccati Equations. This data can be used in a regression f
ramework for which we consider different approximation architectures: poly
nomial approximation\, tensor train decompositions\, and deep neural netwo
rks. Finally\, I will address the role of synthetic data in the framework
of physics-informed neural networks.\nMartin Larsson\nTitle: Minimum curva
ture flow and martingale exit times\nAbstract: We study the following ques
tion: What is the largest deterministic amount of time T* that a suitably
normalized martingale X can be kept inside a convex body K in d-dimensiona
l Euclidean space? We show\, in a viscosity framework\, that T* equals the
time it takes for the relative boundary of K to reach X(0) as it undergoe
s a geometric flow that we call (positive) minimum curvature flow. This re
sult has close links to the literature on stochastic and game representati
ons of geometric flows. Moreover\, the minimum curvature flow can be viewe
d as an arrival time version of the Ambrosio-Soner codimension-(d-1) mean
curvature flow of the 1-skeleton of K. We present preliminary sampling-bas
ed numerical approximations to the solution of the corresponding PDE. The
numerical part is work in progress. This work is based on a collaboration
with Camilo Garcia Trillos\, Johannes Ruf\, and Yufei Zhang.\nAthena Picar
elli\nTitle: A deep solver for BSDEs with jumps\nAbstract: The aim of this
work is to propose an extension of the Deep BSDE solver by Han\, E\, Jent
zen (2017) to the case of FBSDEs with jumps. As in the aforementioned solv
er\, starting from a discretized version of the BSDE and parametrizing the
(high dimensional) control processes by means of a family of ANNs\, the B
SDE is viewed as model-based reinforcement learning problem and the ANN pa
rameters are fitted so as to minimize a prescribed loss function. We take
into account both finite and infinite jump activity\, introducing in the l
atest case\, an approximation with finitely many jumps of the forward proc
ess. (joint work with A. Gnoatto and M. Patacca)\nChristoph Reisinger\nTit
le: Complexity of neural network approximations to parametric and non-para
metric parabolic PDEs\nAbstract: In the first part of the talk\, we discus
s theoretical results which ascertain that deep neural networks can approx
imate the solutions of parabolic PDEs without the curse of dimensionality.
That is to say\, under certain technical assumptions\, there exists a fam
ily of feedforward neural networks such that the complexity\, measured b
y the number of hyper parameters\, grows only polynomially in the dimens
ion of the PDE and the reciprocal of the required accuracy. This part is b
ased on joint work with Yufei Zhang. In the second part\, we study multile
vel neural networks for approximating the parametric dependence of PDE sol
utions. This essentially requires learning a function from computationally
expensive samples. To reduce the complexity\, we construct multilevel est
imators using a hierarchy of finite difference approximations to the PDE o
n refined meshes. We provide a general framework that demonstrates that th
e complexity can be reduced by orders of magnitude under reasonable assump
tions\, and give a numerical illustration with Black-Scholes-type PDEs and
random feature neural networks. This part is based on joint work with Fil
ippo de Angelis and Mike Giles.\nYuri Saporito\nTitle: Gradient Boosting f
or Solving PDEs\nAbstract: Several Deep Learning methods have been success
fully applied to solve several PDEs with many interesting complexities (hi
gh-dimensional\, non-linear\, system of PDEs\, etc). However\, DL usually
lacks proper statistical guarantees and convergence is usually just verifi
ed in practice. In this talk\, we propose a Gradient Boosting method to so
lve a class of PDEs. Although still preliminary\, there is some hope to de
rive proper statistical analysis of the method. Numerical implementations
and examples will be discussed.\nCristopher Salvi\nTitle: Signature kernel
methods for path-dependent PDEs\nAbstract: In this talk we will present a
kernel framework for solving linear and non-linear path-dependent PDEs (P
PDEs) leveraging a recently introduced class of kernels indexed on pathspa
ce\, called signature kernels. The proposed method solves an optimal recov
ery problem by approximating the solution of a PPDE with an element of min
imal norm in the (signature) reproducing kernel Hilbert space (RKHS) const
rained to satisfy the PPDE at a finite collection of collocation points. I
n the linear case\, by the representer theorem\, it can be shown that the
optimisation has a unique\, analytic solution expressed entirely in terms
of simple linear algebra operations. Furthermore\, this method can be exte
nded to a probabilistic Bayesian framework allowing to represents epistemi
c uncertainty over the approximated solution\; this is done by positing a
Gaussian process prior for the solution and constructing a posterior Gauss
ian measure by conditioning on the PPDE being satisfied at a finite collec
tion of points\, with mean function agreeing with the solution of the opti
mal recovery problem. In the non-linear case\, the optimal recovery proble
m can be reformulated as a two-level optimisation that can be solved by mi
nimising a quadratic objective subject to nonlinear constraints. Although
the theoretical analysis is still ongoing\, the proposed method comes with
convergence guarantees and is amenable to rigorous error analysis. Finall
y we will discuss some motivating examples from rough volatility and prese
nt some preliminary numerical results on path-dependent heat-type equation
s. This is joint work with Alexandre Pannier.\nMarc Sabate Vidales\nTitle:
Solving path dependent PDEs with LSTMs and paths signatures\nAbstract: Us
ing a combination of recurrent neural networks and signature methods from
the rough paths theory we design efficient algorithms for solving parametr
ic families of path dependent partial differential equations (PPDEs) that
arise in pricing and hedging of path-dependent derivatives or from use of
non-Markovian model\, such as rough volatility models in Jacquier and Oumg
ari\, 2019. The solutions of PPDEs are functions of time\, a continuous pa
th (the asset price history) and model parameters. As the domain of the so
lution is infinite dimensional many recently developed deep learning techn
iques for solving PDEs do not apply. Similarly as in Vidales et al. 2018\,
we identify the objective function used to learn the PPDE by using martin
gale representation theorem. As a result we can de-bias and provide confid
ence intervals for then neural network-based algorithm. We validate our al
gorithm using classical models for pricing lookback and auto-callable opti
ons and report errors for approximating both prices and hedging strategies
.\n\nYufei Zhang\nTitle: Provably convergent policy gradient methods for c
ontinuous-time stochastic control\nAbstract: Recently\, policy gradient me
thods for stochastic control have attracted substantial research interests
. Much of the attention and success\, however\, has been for the discrete-
time setting. A provably convergent policy gradient method for general con
tinuous space-time stochastic control problems has been elusive. This talk
proposes a proximal gradient algorithm for finite-time horizon stochastic
control problems with controlled drift\, and nonsmooth nonconvex objectiv
e functions. We prove under suitable conditions that the algorithm converg
es linearly to a stationary point of the control problem. We then discuss
a PDE-based\, momentum accelerated implementation of the proposed algorith
m. Numerical experiments for high-dimensional mean-field control problems
are presented\, which reveal that our algorithm captures important structu
res of the optimal policy and achieves a robust performance with respect t
o parameter perturbation. This is joint work with Christoph Reisinger and
Wolfgang Stockinger.
URL:https://www.imperial.ac.uk/events/148795/machine-learning-for-pdes/
DTSTART;TZID=Europe/London:20220906T133000
DTEND;TZID=Europe/London:20220908T124500
LOCATION:LSE\, Old Building\, Room OLD.4.10\, 4th floor\, Houghton Street\,
London\, WC2A 2AE\, United Kingdom
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