Patrick Cheridito

Davide Gallon, Philippe von Wurstemberger, Patrick Cheridito et al.

We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems. We learn the joint distribution of PDE parameters and solutions via a diffusion process in a latent space of scaled spectral representations, where Gaussian noise corresponds to functions with controlled regularity. This spectral formulation enables significant dimensionality reduction compared to grid-based diffusion models and ensures that the induced process in function space remains within a class of functions for which the PDE operators are well defined. Building on diffusion posterior sampling, we enforce physics-informed constraints and measurement conditions during inference, applying Adam-based updates at each diffusion step. We evaluate the proposed approach on Poisson, Helmholtz, and incompressible Navier--Stokes equations, demonstrating improved accuracy and computational efficiency compared with existing diffusion-based PDE solvers, which are state of the art for sparse observations. Code is available at https://github.com/deeplearningmethods/PISD.

Pierfrancesco Beneventano, Patrick Cheridito, Robin Graeber et al.

The purpose of this article is to develop a machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DNNs have the expressive power to overcome the curse of dimensionality in the approximation of a large class of functions. More precisely, we prove that these functions can be approximated by DNNs on compact sets such that the number of parameters necessary to represent the approximating DNNs grows at most polynomially in the reciprocal $1/\varepsilon$ of the prescribed approximation error $\varepsilon>0$ and in the input dimension $d\in\mathbb N$. To this end, we introduce certain approximation spaces, consisting of sequences of functions that can be efficiently approximated by DNNs. We then establish closure properties which we combine with known and new bounds on the number of parameters necessary to approximate locally Lipschitz continuous functions, maximum functions, and product functions by DNNs. The main result of this article demonstrates that DNNs have sufficient expressive power to approximate, without the curse of dimensionality, certain sequences of functions which can be constructed by means of a finite number of compositions using locally Lipschitz continuous functions, maxima, and products.

Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek

Dynamical systems theory has recently been applied in optimization to prove that gradient descent algorithms bypass so-called strict saddle points of the loss function. However, in many modern machine learning applications, the required regularity conditions are not satisfied. In this paper, we prove a variant of the relevant dynamical systems result, a center-stable manifold theorem, in which we relax some of the regularity requirements. We explore its relevance for various machine learning tasks, with a particular focus on shallow rectified linear unit (ReLU) and leaky ReLU networks with scalar input. Building on a detailed examination of critical points of the square integral loss function for shallow ReLU and leaky ReLU networks relative to an affine target function, we show that gradient descent circumvents most saddle points. Furthermore, we prove convergence to global minima under favourable initialization conditions, quantified by an explicit threshold on the limiting loss.

Aleksey Minabutdinov, Patrick Cheridito

We introduce a novel deep learning algorithm for computing convex conjugates of differentiable convex functions, a fundamental operation in convex analysis with various applications in different fields such as optimization, control theory, physics and economics. While traditional numerical methods suffer from the curse of dimensionality and become computationally intractable in high dimensions, more recent neural network-based approaches scale better, but have mostly been studied with the aim of solving optimal transport problems and require the solution of complicated optimization or max-min problems. Using an implicit Fenchel formulation of convex conjugation, our approach facilitates an efficient gradient-based framework for the minimization of approximation errors and, as a byproduct, also provides a posteriori error estimates for the approximation quality. Numerical experiments demonstrate our method's ability to deliver accurate results across different high-dimensional examples. Moreover, by employing symbolic regression with Kolmogorov--Arnold networks, it is able to obtain the exact convex conjugates of specific convex functions.

Jean-Loup Dupret, Davide Gallon, Patrick Cheridito

In this paper, we introduce INEUS, a meshfree iterative neural solver for partial integro-differential equations (PIDEs). The method replaces the explicit evaluation of nonlocal jump integrals with single-jump sampling and reformulates PIDE solving as a sequence of recursive regression problems. Like Physics-Informed Neural Networks (PINNs), INEUS learns global solutions over the entire space-time domain, yet it offers a more efficient treatment of nonlocal terms and avoids the computationally expensive differentiation of full PIDE residuals. These features make INEUS particularly well suited for high-dimensional PDEs and PIDEs. Supported by a contraction-based convergence proof for linear PIDEs, our numerical experiments show that INEUS delivers accurate and scalable solutions for various high-dimensional linear and nonlinear examples.

Patrick Cheridito, Jean-Loup Dupret, Donatien Hainaut

In this paper, we introduce a model-based deep-learning approach to solve finite-horizon continuous-time stochastic control problems with jumps. We iteratively train two neural networks: one to represent the optimal policy and the other to approximate the value function. Leveraging a continuous-time version of the dynamic programming principle, we derive two different training objectives based on the Hamilton-Jacobi-Bellman equation, ensuring that the networks capture the underlying stochastic dynamics. Empirical evaluations on different problems illustrate the accuracy and scalability of our approach, demonstrating its effectiveness in solving complex, high-dimensional stochastic control tasks.

Gabriele Visentin, Patrick Cheridito

We present a novel method for efficiently computing optimal transport maps and Wasserstein barycenters in high-dimensional spaces. Our approach uses conditional normalizing flows to approximate the input distributions as invertible pushforward transformations from a common latent space. This makes it possible to directly solve the primal problem using gradient-based minimization of the transport cost, unlike previous methods that rely on dual formulations and complex adversarial optimization. We show how this approach can be extended to compute Wasserstein barycenters by solving a conditional variance minimization problem. A key advantage of our conditional architecture is that it enables the computation of barycenters for hundreds of input distributions, which was computationally infeasible with previous methods. Our numerical experiments illustrate that our approach yields accurate results across various high-dimensional tasks and compares favorably with previous state-of-the-art methods.

Gabriele Visentin, Patrick Cheridito

In this paper, we show that interventionally robust optimization problems in causal models are continuous under the $G$-causal Wasserstein distance, but may be discontinuous under the standard Wasserstein distance. This highlights the importance of using generative models that respect the causal structure when augmenting data for such tasks. To this end, we propose a new normalizing flow architecture that satisfies a universal approximation property for causal structural models and can be efficiently trained to minimize the $G$-causal Wasserstein distance. Empirically, we demonstrate that our model outperforms standard (non-causal) generative models in data augmentation for causal regression and mean-variance portfolio optimization in causal factor models.

Patrick Cheridito, Balint Gersey

Theoretically, the conditional expectation of a square-integrable random variable $Y$ given a $d$-dimensional random vector $X$ can be obtained by minimizing the mean squared distance between $Y$ and $f(X)$ over all Borel measurable functions $f \colon \mathbb{R}^d \to \mathbb{R}$. However, in many applications this minimization problem cannot be solved exactly, and instead, a numerical method which computes an approximate minimum over a suitable subfamily of Borel functions has to be used. The quality of the result depends on the adequacy of the subfamily and the performance of the numerical method. In this paper, we derive an expected value representation of the minimal mean squared distance which in many applications can efficiently be approximated with a standard Monte Carlo average. This enables us to provide guarantees for the accuracy of any numerical approximation of a given conditional expectation. We illustrate the method by assessing the quality of approximate conditional expectations obtained by linear, polynomial and neural network regression in different concrete examples.

Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek

In this paper, we analyze the landscape of the true loss of neural networks with one hidden layer and ReLU, leaky ReLU, or quadratic activation. In all three cases, we provide a complete classification of the critical points in the case where the target function is affine and one-dimensional. In particular, we show that there exist no local maxima and clarify the structure of saddle points. Moreover, we prove that non-global local minima can only be caused by `dead' ReLU neurons. In particular, they do not appear in the case of leaky ReLU or quadratic activation. Our approach is of a combinatorial nature and builds on a careful analysis of the different types of hidden neurons that can occur.

Patrick Cheridito, Arnulf Jentzen, Adrian Riekert et al.

Gradient descent optimization algorithms are the standard ingredients that are used to train artificial neural networks (ANNs). Even though a huge number of numerical simulations indicate that gradient descent optimization methods do indeed convergence in the training of ANNs, until today there is no rigorous theoretical analysis which proves (or disproves) this conjecture. In particular, even in the case of the most basic variant of gradient descent optimization algorithms, the plain vanilla gradient descent method, it remains an open problem to prove or disprove the conjecture that gradient descent converges in the training of ANNs. In this article we solve this problem in the special situation where the target function under consideration is a constant function. More specifically, in the case of constant target functions we prove in the training of rectified fully-connected feedforward ANNs with one-hidden layer that the risk function of the gradient descent method does indeed converge to zero. Our mathematical analysis strongly exploits the property that the rectifier function is the activation function used in the considered ANNs. A key contribution of this work is to explicitly specify a Lyapunov function for the gradient flow system of the ANN parameters. This Lyapunov function is the central tool in our convergence proof of the gradient descent method.

Christian Beck, Sebastian Becker, Patrick Cheridito et al.

In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied to a set of simulated noise trajectories, it yields empirical distributions of SPDE solutions, from which functionals like the mean and variance can be estimated. We test the performance of the method on stochastic heat equations with additive and multiplicative noise as well as stochastic Black-Scholes equations with multiplicative noise and Zakai equations from nonlinear filtering theory. In all cases, the proposed algorithm yields accurate results with short runtimes in up to 100 space dimensions.

Patrick Cheridito, Arnulf Jentzen, Florian Rossmannek

Deep neural networks have successfully been trained in various application areas with stochastic gradient descent. However, there exists no rigorous mathematical explanation why this works so well. The training of neural networks with stochastic gradient descent has four different discretization parameters: (i) the network architecture; (ii) the amount of training data; (iii) the number of gradient steps; and (iv) the number of randomly initialized gradient trajectories. While it can be shown that the approximation error converges to zero if all four parameters are sent to infinity in the right order, we demonstrate in this paper that stochastic gradient descent fails to converge for ReLU networks if their depth is much larger than their width and the number of random initializations does not increase to infinity fast enough.

Sebastian Becker, Patrick Cheridito, Arnulf Jentzen et al.

Nowadays many financial derivatives, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlying assets. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work, we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options, such as Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.

Christian Beck, Sebastian Becker, Patrick Cheridito et al.

In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high-dimensional PDEs. We test the method on different examples from physics, stochastic control and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.