The goal of the workshop is to gather exper ts from different areas in this inter-disciplinary field to investigate an d discuss how to harness the power of machine learning techniques to solve high-dimensional\, non-linear partial differential equations (PDEs)\, as well as how to leverage the theory of PDEs to construct better machine lea rning models and study their theoretical properties.

\nPDEs are a do minant modelling paradigm ubiquitous throughout science\, from fluid dynam ics to quantum mechanics\, to calculus of variations and quantitative fina nce. When the PDEs at hand are low dimensional (dim=1\,2\,3\,4) they can g enerally be solved numerically leveraging a large arsenal of techniques de veloped over the last 150 years\, including finite difference and finite e lements methods.

\nNonetheless\, many PDEs arising from complex\, re al world financial-engineering-physical problems are often so high-dimensi onal (sometimes even infinite dimensional) that classical numerical techni ques are either not directly applicable or do not scale to high-resolution computations. Examples of such intractable equations include pricing and hedging with rough volatility price dynamics\, non-Markovian\, path-depend ent stochastic control problems\, and turbulent fluid flow dynamics to be solved on very fine scales.

\nRecent advances in Machine Learning (M L) have enabled for the development of novel computational techniques for tackling PDE-based problems considered unresolvable with classical methods . Physics-informed neural networks\, neural differential equations\, and n eural operators are among the most popular models used to tackle PDE-relat ed problems with deep learning.

\nThe goal of this workshop is to de velop a classification of ML techniques depending on the type of PDE and t o set clear new directions in the design of optimal numerical schemes\, bo th numerically and theoretically (with convergence results). The list of p articipants is designed to maximise inter-disciplinarity and encourage div ersity with experts in different fields\, such as stochastic analysis\, nu merical analysis\, mathematical finance and machine learning.

\nDr A. Jacquier (Imperial College)\, Prof. J. Ruf (LSE) an d Dr C. Salvi (Imperial College).

\nPlease contact the organisers if you are interested in attending the workshop.

\nEPSRC\, LSE\, Imperial.

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\n## Titles and abstracts

\n**Athena Picarelli**

\n**Title**: A deep s
olver for BSDEs with jumps

\n**Abstract**: The aim of
this work is to propose an extension of the Deep BSDE solver by Han\, E\,
Jentzen (2017) to the case of FBSDEs with jumps. As in the aforementioned
solver\, starting from a discretized version of the BSDE and parametrizin
g the (high dimensional) control processes by means of a family of ANNs\,
the BSDE is viewed as model-based reinforcement learning problem and the A
NN parameters are fitted so as to minimize a prescribed loss function. We
take into account both finite and infinite jump activity\, introducing in
the latest case\, an approximation with finitely many jumps of the forward
process. (joint work with A. Gnoatto and M. Patacca)\n

Tuesday 6 th | 13:30-13:45 | Welcome Speech |

< /td>\n | 13:45-14:45 | Anas tasia Borovykh |

14:45-15:45 | Yuri Saporito | |

15:45-16:15 | Coffee Break | |

16:15-17:15 | Yufei Zhang | |

Wedn esday 7th | 10:15-11:15 | Martin Larsson |

11:15-11:45 | M arc Sabate Vidales | |

11:45-12:30 | Sam Cohen | |

12:30-13:45 | Lunch Break | |

13:45-14:45 | Stefania Fresca | |

14:45-15:45 | Christoph Reisinger | |

15:45-16:15 | Coffee Break | |

\n | 16:15-17:15 | Dante Kalise |

19:00- 22:00 | Workshop Dinner (by invitation only) | |

10:00-10:30 | Athena Picarelli | |

10:30-11:30 | Cris Salvi | |

11:30-11:45 | Coffee Break | |

1 1:45-12:45 | Camilo Garcia Trillos |

\n\n

**Anastasia Borovykh**

\n**Title**: On the choice of loss functions and initiali
zations for deep learning-based solvers for PDEs

\n**Abstract: **I
n this talk we will discuss several challenges that arise when solving PDE
s with deep learning-based solvers. We will begin with defining the loss f
unction of a general PDE and discuss how this choice of loss function\, an
d specifically the weighting of the different loss terms\, can impact the
accuracy of the solution. We will show how to choose an optimal weighting
that corresponds to accurate solutions. Next\, we will focus on the approx
imation of the Hamilton-Jacobi-Bellman partial differential equation assoc
iated to optimal stabilization of the NonLinear Quadratic Regular Problem.
It is not obvious that the neural network will converge to the correct so
lution with just any type of initialisation\; this is particularly relevan
t when the solution to the HJB-PDE is non-unique. We will discuss a two-st
ep learning approach where the model is pre-trained on a dataset obtained
from solving a state-dependent Riccati equation and we show that in this w
ay efficient and accurate convergence can still be obtained.

\n

**Sam Cohen**

\n**Title: **Neural Q-learning solution
s to elliptic PDEs

\nAbstract: Solving high-dimensional partial diffe
rential 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. 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 tr
ajectory of an infinite-dimensional ordinary differential equation (ODE) a
s the number of hidden units becomes infinite. For monotone PDE (i.e. thos
e 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 i
nfinite-dimensional ODE\, converges in $L^2$ to the PDE solution as the tr
aining time increases. The numerical performance of the Q-PDE algorithm is
studied for several elliptic PDEs. Based on joint work with Deqing Jiang
and Justin Sirignano.

**Stefania Fresca**

\n<
b>Title: Deep learning-based reduced order models for scientific
applications

\n**Abstract**: The solution of differe
ntial equations by means of full order models (FOMs)\, such as\, e.g.\, th
e finite element method\, entails prohibitive computational costs when it
comes to real-time simulations and multi-query routines. The purpose of re
duced order modeling is to replace FOMs with suitable surrogates\, so-call
ed reduced order models (ROMs)\, characterized by much lower complexity bu
t still able to express

\nthe physical features of the system under i
nvestigation. Conventional ROMs anchored to the assumption of modal linear
superimposition\, such as proper orthogonal decomposition (POD)\, may rev
eal inefficient when dealing with nonlinear time-dependent parametrized PD
Es\, especially for problems featuring coherent structures propagating ove
r time. To enhance ROM efficiency\, 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 nonlinear reduced dynamics are learned in a non-intrusiv
e way by relying on DL algorithms trained on a set of FOM snapshots\, obta
ined for different parameter values. Furthermore\, in case of large-scale
FOMs\, a former dimensionality reduction on FOM snapshots through POD enab
les to speed-up training times and to substantially decrease the network c
omplexity. Accuracy and efficiency of the DL-ROM technique are assessed in
different scientific applications aiming at solving parametrized PDE prob
lems\, e.g.\, in cardiac electrophysiology\, computational mechanics and f
luid dynamics\, possibly accounting for fluid-structure interaction effect
s\, 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 mathe
matical foundations\, error bounds are derived for the approximation of no
nlinear operators by means of CNNs. The resulting error estimates provide
a clear interpretation of the hyperparameters defining the neural network
architecture.

**Camilo Garcia Trillos**

\n**
Title**: Neural network approximation of some second order semilinear PD
Es

\n**Abstract**: Since its inception in the early 90s\, the well
-known connection between second-order semi-linear PDEs and Markovian BSDE
s has been useful in creating numerical probabilistic methods to solve the
former. Our approach to the solution of these PDEs belongs to a recent st
ream in the literature that uses neural networks together with the BSDE co
nnection to define numerical methods that are robust and efficient in larg
e dimensions. In contrast with existing works\, our analysis focuses on th
e case where derivatives enter ‘quadratically’ in the semilinear term\
, covering some interesting cases in control theory. In this setting\, we
study both forward and backward-types of neural network based methods. Joi
nt work with Daniel Bussell.

**Dante Kalise**

\n**Title**: Data-driven schemes for Hamilton-Jacobi-Bell
man equations

\n**Abstract**: In this talk I will dis
cuss different computational aspects arising in the construction of data-d
riven schemes for HJB PDEs. First\, I will discuss synthetic data generati
on through representation formulas including Pontryagin’s Maximum Princi
ple and State-dependent Riccati Equations. This data can be used in a regr
ession framework for which we consider different approximation architectur
es: polynomial approximation\, tensor train decompositions\, and deep neur
al networks. Finally\, I will address the role of synthetic data in the fr
amework of physics-informed neural networks.

**Martin Lars
son**

\n**Title**: Minimum curvature flow and martingale e
xit times

\n**Abstract**: 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.

\n

\n

**Ch
ristoph Reisinger**

\n**Title**: Complexity of neural netw
ork approximations to parametric and non-parametric parabolic PDEs<
br />\n**Abstract**: In the first part of the talk\, we discuss t
heoretical results which ascertain that deep neural networks can approxima
te the solutions of parabolic PDEs without the curse of dimensionality. Th
at is to say\, under certain technical assumptions\, there exists a family
of feedforward neural networks such that the complexity\, measured by t
he number of hyper parameters\, grows only polynomially in the dimension
of the PDE and the reciprocal of the required accuracy. This part is base
d on joint work with Yufei Zhang. In the second part\, we study multilevel
neural networks for approximating the parametric dependence of PDE soluti
ons. This essentially requires learning a function from computationally ex
pensive samples. To reduce the complexity\, we construct multilevel estima
tors using a hierarchy of finite difference approximations to the PDE on r
efined meshes. We provide a general framework that demonstrates that the c
omplexity can be reduced by orders of magnitude under reasonable assumptio
ns\, and give a numerical illustration with Black-Scholes-type PDEs and ra
ndom feature neural networks. This part is based on joint work with Filipp
o de Angelis and Mike Giles.

**Yuri Saporito**

\n**Title**: Gradient Boosting for Solving PDEs

\n**Abstract**: Several Deep Learning methods have been successfully applied to solve s
everal PDEs with many interesting complexities (high-dimensional\, non-lin
ear\, system of PDEs\, etc). However\, DL usually lacks proper statistical
guarantees and convergence is usually just verified in practice. In this
talk\, we propose a Gradient Boosting method to solve a class of PDEs. Alt
hough still preliminary\, there is some hope to derive proper statistical
analysis of the method. Numerical implementations and examples will be dis
cussed.

**Cristopher Salvi**

\n**Title: **Signat
ure kernel methods for path-dependent PDEs

\n**Abstract**: In this
talk we will present a kernel framework for solving linear and non-
linear path-dependent PDEs (PPDEs) leveraging a recently introduced class
of kernels indexed on pathspace\, called signature kernels. The proposed m
ethod solves an optimal recovery problem by approximating the solution of
a PPDE with an element of minimal norm in the (signature) reproducing kern
el Hilbert space (RKHS) constrained to satisfy the PPDE at a finite collec
tion of collocation points. In the linear case\, by the representer theore
m\, it can be shown that the optimisation has a unique\, analytic solution
expressed entirely in terms of simple linear algebra operations. Furtherm
ore\, this method can be extended to a probabilistic Bayesian framework al
lowing to represents epistemic uncertainty over the approximated solution\
; this is done by positing a Gaussian process prior for the solution and c
onstructing a posterior Gaussian measure by conditioning on the PPDE being
satisfied at a finite collection of points\, with mean function agreeing
with the solution of the optimal recovery problem. In the non-linear case\
, the optimal recovery problem can be reformulated as a two-level optimisa
tion that can be solved by minimising a quadratic objective subject to non
linear constraints. Although the theoretical analysis is still ongoing\, t
he proposed method comes with convergence guarantees and is amenable to ri
gorous error analysis. Finally we will discuss some motivating examples fr
om rough volatility and present some preliminary numerical results on path
-dependent heat-type equations. This is joint work with Alexandre Pannier.

**Marc Sabate Vidales**

\n**Title**: Solving pa
th dependent PDEs with LSTMs and paths signatures

\n**Abstract**:
Using a combination of recurrent neural networks and signature methods fro
m the rough paths theory we design efficient algorithms for solving parame
tric families of path dependent partial differential equations (PPDEs) tha
t arise in pricing and hedging of path-dependent derivatives or from use o
f non-Markovian model\, such as rough volatility models in Jacquier and Ou
mgari\, 2019. The solutions of PPDEs are functions of time\, a continuous
path (the asset price history) and model parameters. As the domain of the
solution is infinite dimensional many recently developed deep learning tec
hniques 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 mart
ingale representation theorem. As a result we can de-bias and provide conf
idence intervals for then neural network-based algorithm. We validate our
algorithm using classical models for pricing lookback and auto-callable op
tions and report errors for approximating both prices and hedging strategi
es.

\n

**Yufei Zhang**

\n**Title**: P
rovably convergent policy gradient methods for continuous-time stochastic
control

\n**Abstract**: 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.