Nonlinear & Nonlocal High-Order PDE Systems

• Nonlinear and nonlocal high order PDE systems are evolution equations characterized by nonlinear dependencies, integral operators inducing nonlocality, and higher-order derivative terms.
• They employ advanced numerical schemes, such as probabilistic Monte Carlo and deep BSDE methods, to tackle high-dimensional challenges and ensure convergence.
• Applications in stochastic control, jump-diffusion finance, and reaction-diffusion models highlight their practical significance in modeling complex physical and biological phenomena.

A nonlinear and nonlocal high order system of partial differential equations (PDEs) encompasses evolution equations where (1) the dependence on the solution and its derivatives is nonlinear, (2) integral operators introduce nonlocality—meaning the value at a point depends on the solution elsewhere—and (3) higher-order spatial and temporal derivatives or integro-differential components are present. Such systems arise in applications including stochastic control, mean-field games, jump-diffusion financial models, and nonlocal reaction-diffusion problems in physics and biology. Notable mathematical challenges include singular Lévy measures (leading to infinite activity jumps), unbounded nonlocal operators, and high-dimensional state spaces.

1. Mathematical Formulation and Examples

A general prototype is the nonlinear parabolic integro-differential equation: $$\partial_t u(t,x) + \mathcal{L}[u](t,x) + \mathcal{J}[u](t,x) + f\left(t,x,u,\nabla_x u, D^2_x u, \mathcal{B}[u]\right) = 0,$$ where $u:[0,T]\times D\rightarrow\mathbb{R}$ (with $D\subseteq\mathbb{R}^d$), $\mathcal{L}$ is a local elliptic operator (e.g., diffusion and drift), $\mathcal{J}$ is a nonlocal Lévy-type operator or general nonlocal interaction, $f$ is a possibly nonlinear reaction or control term, and $\mathcal{B}$ represents additional nonlocal couplings.

Fully Nonlinear Nonlocal HJBI PDEs: The Hamilton-Jacobi-Bellman-Isaacs (HJBI) framework features nonlinearity through an inf-sup over control sets, and the generator includes both local and nonlocal jump terms (Fahim, 2010): $$- \partial_t u - L^x u - F(t,x,u,Du,D^2u,u(t,\cdot)) = 0, \quad u(T,\cdot) = g,$$ with $F$ defined as an inf-sup of local and nonlocal integral parts parameterized by control indices.

Nonlocal Reaction-Diffusion Systems: These model spatial interactions at a distance via integrals against kernel measures or through interaction nonlinearities, e.g.,

$$\frac{\partial u}{\partial t} = \mu(x)\cdot\nabla_x u + \frac12 \mathrm{Tr}[\sigma(x)\sigma(x)^\top D_x^2 u] + \int_D f(t,x,x',u(t,x),u(t,x'))\,\nu_x(dx'),$$

where $u:[0,T]\times D\rightarrow\mathbb{R}$0 encodes nonlocal nonlinear interactions and $u:[0,T]\times D\rightarrow\mathbb{R}$1 prescribes interaction measures (Boussange et al., 2022).

2. Stochastic Representations and Viscosity Solutions

Viscosity solution theory underpins the well-posedness of nonlinear nonlocal PDEs, especially when solutions may be nonsmooth due to singularities or degeneracy. For equations associated with Lévy-driven stochastic processes, the nonlocal integro-differential operator admits a Feynman-Kac-type probabilistic representation—using forward-backward stochastic differential equations (FBSDEs) driven by Brownian motion and Poisson random measures. This is explicitly exploited in the context of unbounded/infinite-activity jump processes (Jakobsen et al., 2024): $u:[0,T]\times D\rightarrow\mathbb{R}$2 This suggests that the nonlinear, nonlocal PDE solution can be constructed as the first component of the FBSDE, providing a powerful route for both analysis and numerical approximation.

3. Truncation and Approximation of Lévy-type Nonlocal Terms

Nonlocal operators arising from singular Lévy measures demand specific handling due to the possible divergence of small-jump (infinite activity) contributions. A standard technique is truncation: one replaces the singular integral $u:[0,T]\times D\rightarrow\mathbb{R}$3 by integrating over $u:[0,T]\times D\rightarrow\mathbb{R}$4 (for $u:[0,T]\times D\rightarrow\mathbb{R}$5), and either neglects or compensates the small-jump component.

In Monte Carlo Quadrature (MCQ), as in (Fahim, 2010), the truncated operator is simulated directly: $u:[0,T]\times D\rightarrow\mathbb{R}$6 with its expectation approximated by compound-Poisson processes. Error analysis demonstrates explicit bounds in terms of $u:[0,T]\times D\rightarrow\mathbb{R}$7 bias and truncation residuals $u:[0,T]\times D\rightarrow\mathbb{R}$8, establishing convergence guarantees provided the truncation threshold and time steps are balanced appropriately.

For FBSDE-based approaches (Jakobsen et al., 2024), small-jump truncation at $u:[0,T]\times D\rightarrow\mathbb{R}$9 is combined with a Gaussian correction for retained jump activity (if $D\subseteq\mathbb{R}^d$0), ensuring accurate weak approximation with control over the error rate as $D\subseteq\mathbb{R}^d$1.

4. Numerical Schemes: Monte Carlo and Machine Learning

Traditional grid-based methods are intractable for high-dimensional, nonlinear, and nonlocal systems due to exponential scaling. Probabilistic and mesh-free methods dominate current research, with two main classes highlighted:

Probabilistic Monte Carlo Schemes: These include time-discretized, backward-iteration schemes based on the Euler–Poisson process, Hermite interpolation for derivatives, and MCQ for nonlocal terms (Fahim, 2010). Convergence theorems guarantee that with appropriate time-and-jump-step selection, the discrete scheme converges locally uniformly to the viscosity solution.

Deep BSDE Approaches: Machine learning-enabled regression is used for the backward components of FBSDEs, fitting neural networks to represent solution components (the value, gradient, jump increment, and Gaussian correction terms) at each time step (Jakobsen et al., 2024). The method admits rigorous convergence analysis, circumventing the curse of dimensionality and handling unbounded nonlocal operators arising from infinite-activity Lévy processes.

Deep-Splitting and Picard Iteration Methods: For PDEs with Neumann boundary conditions or nonlocal reaction terms, time-splitting is combined with neural network surrogates for the solution, leveraging splitting schemes for local vs. nonlocal terms and stochastic gradient descent for optimization (Boussange et al., 2022). The multilevel Picard (MLP) method extends this to recursive single-point evaluation, with provable error and complexity bounds polynomial in $D\subseteq\mathbb{R}^d$2 and dimension.

5. Error Analysis and Convergence Guarantees

Rigorous rates of convergence and error bounds are central to the numerical analysis of nonlinear and nonlocal PDEs:

• In concave/convex cases, sharper error rates are available. For the MCQ truncation scheme, if step size $D\subseteq\mathbb{R}^d$3 and truncation $D\subseteq\mathbb{R}^d$4 are chosen such that $D\subseteq\mathbb{R}^d$5 and $D\subseteq\mathbb{R}^d$6, the solution error is $D\subseteq\mathbb{R}^d$7 (Fahim, 2010).
• In deep BSDE approaches, the overall error combines discretization, jump truncation, and neural network approximation errors. Each term can be made arbitrarily small by step refinement and network size increase, yielding provable approximation to the true viscosity solution (with known dependence on dimension and time discretization) (Jakobsen et al., 2024).
• For both deep-splitting and MLP strategies, numerical tests and theory confirm polynomial scaling in both tolerance and dimension, extending practical capability to $D\subseteq\mathbb{R}^d$8 (and thousands in some prior works), thus overcoming the curse of dimensionality (Boussange et al., 2022).

6. Applications and Numerical Results

Nonlinear and nonlocal high order systems of PDEs appear in a broad range of complex models:

• Stochastic control and game theory: HJBI equations with nonlocal terms model optimal control under jump-diffusion and mean-field game settings (Fahim, 2010, Jakobsen et al., 2024).
• Physics and biology: Nonlocal reaction-diffusion systems—such as Allen–Cahn, Fisher–KPP, sine–Gordon, and replicator-mutator equations—feature integral interactions modeling spatial effects in population dynamics, phase transitions, and pattern formation (Boussange et al., 2022).
• Finance: Jump-diffusion models for pricing derivatives under risk of sudden events necessitate unbounded Lévy nonlocal operators (Fahim, 2010, Jakobsen et al., 2024).

Numerical experiments have demonstrated the efficacy of learning-based schemes in up to $D\subseteq\mathbb{R}^d$9 dimensions for nonlocal reaction-diffusion and jump-diffusion equations, with relative $\mathcal{L}$0-errors of order $\mathcal{L}$1–$\mathcal{L}$2 and runtimes under one minute on standard hardware (Boussange et al., 2022). Machine learning approaches maintain accuracy in high dimensions and for complex nonlocal nonlinearities due to their mesh-free, expectation-based sampling strategies.

7. Overcoming the Curse of Dimensionality and Future Directions

All current leading approaches fundamentally avoid exponential complexity by replacing spatial grids with SDE- or FBSDE-based sampling, and by utilizing stochastic optimization in functional spaces. Both MCQ and ML schemes leverage universal approximation capabilities and polynomial cost scaling in both dimension and tolerance (Jakobsen et al., 2024, Boussange et al., 2022). Open frontiers include improved variance reduction, adaptive jump-truncation, and analysis of regularity and approximation-theoretic properties of nonlocal operators in high order nonlinear regimes.

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References:

• "A Probabilistic Scheme for Fully Nonlinear Nonlocal Parabolic PDEs with singular Lévy measures" (Fahim, 2010)
• "A Deep BSDE approximation of nonlinear integro-PDEs with unbounded nonlocal operators" (Jakobsen et al., 2024)
• "Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions" (Boussange et al., 2022)