arxiv: 2605.06281 · v1 · submitted 2026-05-07 · 💻 cs.LG · cs.NA· math.NA· q-fin.CP

Recognition: unknown

INEUS: Iterative Neural Solver for High-Dimensional PIDEs

Jean-Loup Dupret , Davide Gallon , Patrick Cheridito

Authors on Pith no claims yet

Pith reviewed 2026-05-08 13:06 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NAq-fin.CP

keywords neural solverpartial integro-differential equationshigh-dimensional PDEsmeshfree methodsjump integralsregressioniterative methods

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The pith

INEUS replaces explicit nonlocal integrals in PIDEs with single-jump sampling and solves them as recursive regression problems using neural networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents INEUS, a new method to solve partial integro-differential equations in high dimensions without meshes. It avoids computing full jump integrals by sampling just one jump per iteration and breaks the problem into a series of simpler regression tasks that a neural net learns sequentially. This setup learns the solution over the whole domain at once and comes with a mathematical proof that it converges for linear cases. The approach is more efficient than standard physics-informed networks for problems involving jumps and scales better to many dimensions. Readers should care because many models in finance, physics, and biology involve such equations that become intractable in high dimensions with traditional methods.

Core claim

INEUS is a meshfree iterative neural solver for PIDEs that reformulates the equation solving as a sequence of recursive regression problems by replacing the explicit evaluation of nonlocal jump integrals with single-jump sampling. Like PINNs, it learns global solutions over the space-time domain but treats nonlocal terms more efficiently and avoids differentiating full PIDE residuals. A contraction-based convergence proof supports its use for linear PIDEs, and experiments confirm accurate solutions for both linear and nonlinear high-dimensional cases.

What carries the argument

Single-jump sampling that substitutes for full nonlocal integral evaluation, enabling the reformulation of PIDE solving into recursive regression problems solved iteratively by neural networks.

If this is right

Accurate and scalable solutions are obtained for various high-dimensional linear and nonlinear PIDEs.
Nonlocal terms are treated more efficiently without needing to differentiate full residuals.
Convergence is guaranteed for linear PIDEs through a contraction-based proof.
The method learns solutions globally over the entire space-time domain.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Applications in areas like option pricing under jump-diffusion models could benefit from this scalability in high dimensions.
Extensions to other types of nonlocal equations might be possible by adapting the sampling strategy.
Combining this with other neural PDE solvers could lead to hybrid methods for mixed local-nonlocal problems.

Load-bearing premise

Single-jump sampling can replace explicit nonlocal integral evaluation while preserving accuracy and allowing the recursive regressions to converge to the true solution.

What would settle it

Running INEUS on a high-dimensional linear PIDE with a known closed-form solution and observing whether the neural approximation converges to it as the number of iterations increases, consistent with the contraction proof.

Figures

Figures reproduced from arXiv: 2605.06281 by Davide Gallon, Jean-Loup Dupret, Patrick Cheridito.

**Figure 1.** Figure 1: displays the INEUS approximation for this problem in dimension d = 100. In this highdimensional setting with jumps, standard PINNs become computationally intractable, whereas INEUS still provides an accurate approximation of the solution view at source ↗

**Figure 2.** Figure 2: Left panel: comparison of MAE (blue) and runtime in seconds (green) of INEUS (solid lines) and PINN (dashed lines) as functions of the epoch k. Right panel: comparison of INEUS, PINN, and deep BSDE with jumps as functions of runtime (in seconds). Both panels correspond to a 10-dimensional linear PIDE (32) view at source ↗

**Figure 3.** Figure 3: PIDE solution u(0, x) for x = (x1, 0, . . . , 0) with x1 ∈ [−1.5, 1.5] (left) and u(t, x) for t ∈ [0, 1] and x = 0100 (right) for a 100-dimensional nonlinear PIDE (34). Orange dotted lines: numerical results of INEUS with ±1 standard deviation given by orange shaded area. Blue lines: reference MC solution (35) view at source ↗

**Figure 4.** Figure 4: Left panel: comparison of MAE (blue) and runtime in seconds (green) of INEUS (solid lines) and PINN (dashed lines) as functions of the epoch k. Right panel: comparison of INEUS, PINN, and deep BSDE with jumps as functions of runtime (seconds). Both panels correspond to a 10-dimensional nonlinear PIDE (34). Nonlinear Black–Scholes PIDE. We extend the d-dimensional nonlinear Black–Scholes PDE with default ri… view at source ↗

**Figure 5.** Figure 5: INEUS for the 100-dimensional nonlinear Black–Scholes equation (36), without jumps (upper row) and with jumps (lower row). The left panels show the training and test losses over 400 epochs, while the right panels show the prediction trajectory of uθ(0, x0) with x0 = 100 · 1100. The no-jump estimate is compared with the Feynman–Kac reference value 57.300, while the jump estimate is compared with the Picard … view at source ↗

**Figure 6.** Figure 6: DGM architecture of the neural network uθ with L = 3 (i.e. 4 hidden layers). DGM network Each DGM layer of the network uθ in view at source ↗

**Figure 7.** Figure 7: Comparison of INEUS using the Feynman–Kac operator Th (solid line) and the relaxed operator Teh (dotted line, K = 2) in terms of MAE (blue) and runtime in seconds (green), as functions of the number of epochs k for the linear PIDE (32). Finally, we consider the linear PDE (32) without jumps (λ = 0). As shown in view at source ↗

**Figure 8.** Figure 8: Comparison of MAE (blue) and runtime in seconds (green) of INEUS (solid lines) and PINNs (dashed lines) for a 100-dimensional linear PDE (32) without jumps (λ = 0). Nonlinear Hamilton–Jacobi-Bellman PIDE. The parameters for the PIDE (34) are: T = 1, η = 1, f(t, x) = 2, λ = 0.5, µJ = 0d, ΣJ = 0.2Id, and test set uniformly sampled from [0, 1] × [−1.5, 1.5]d view at source ↗

**Figure 9.** Figure 9: Comparison of MAE (blue) and runtime in seconds (green) of INEUS (solid lines) and PINNs (dashed lines) for a 100-dimensional nonlinear PDE (34) without jumps (λ = 0). Linear Black–Scholes PIDE. We consider a financial derivative on a basket of d underlying stocks, whose Black–Scholes pricing PIDE is given on DT = [0, T) × R d + by ∂tu(t, x) + (r − κ ⊙ λ) ⊙ x · ∇xu(t, x) + 1 2 Tr[diag(x)σΣσ ⊤ diag(x)∇ 2 xu… view at source ↗

**Figure 10.** Figure 10: PIDE solution u(0, x) for x = (x1, K, . . . , K) with x1 ∈ [0, 100] (left) and u(t, x) for t ∈ [0, 1] and x = K120 (right) for a 20-dimensional linear Black–Scholes PIDE (49). Orange dotted lines: numerical results of INEUS with ±1 standard deviation given by orange shaded area. Blue lines: reference MC solution obtained from the Feynman–Kac formula (14) with ±1 standard deviation given by blue shaded area view at source ↗

**Figure 11.** Figure 11: Left panel: comparison of MAE (blue) and runtime in seconds (green) of INEUS (solid lines) and PINN (dashed lines) as functions of the epoch k. Right panel: comparison of INEUS, PINN, and deep BSDE with jumps as functions of runtime. Both panels correspond to a 10-dimensional linear Black–Scholes PIDE (49) view at source ↗

**Figure 12.** Figure 12: Comparison of MAE (blue) and runtime in seconds (green) of INEUS (solid lines) and PINNs (dashed lines) for a 100-dimensional linear Black–Scholes PDE (49) without jumps (λ = 0). 26 view at source ↗

**Figure 13.** Figure 13: Learned solution uθ(t, Xt) of the Black–Scholes nonlinear equation (36) without jumps (left, λ = 0) and with jumps (right, λ = 0.1). In both panels, the solution is evaluated along 50 independent geometric Brownian motion sample paths starting from x0 = 100·1100. In the no-jump setting (left), trajectories start from uθ(0, x0) ≈ 57.3 and spread as the diffusion variance grows, consistent with the expected… view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

INEUS turns PIDEs into iterative regressions with single-jump sampling and proves contraction for the linear case, but the sampling variance is not bounded against the exact-operator proof.

read the letter

INEUS recasts high-dimensional PIDEs as a sequence of regression problems where the nonlocal jump integral is replaced by drawing one jump sample per point. This is the main novelty relative to standard PINN-style methods, and it avoids having to differentiate the full residual that includes the integral term. The contraction argument for linear PIDEs gives the approach a bit of theoretical grounding that many neural solvers lack, and the reported experiments cover both linear and nonlinear examples at dimensions where grid methods become impractical. The numerical results show reasonable accuracy and scaling, which is the practical evidence the paper supplies. That part is concrete enough to evaluate on its own terms. The soft spot is the mismatch between the proof and the algorithm. The contraction is shown for the exact integral operator, yet the implemented method uses single-jump Monte Carlo whose variance stays positive and grows with dimension. No error propagation bound or modified contraction statement is given to show that the stochastic perturbation still lets the iterates converge to the true solution. For the nonlinear cases there is no proof at all, so those claims rest solely on the experiments. A reader working on high-dimensional PIDEs in finance or nonlocal physics would find the method worth testing, especially if they already use neural approaches and need something lighter on the integral term. The paper is clear enough on the implementation to be sent to referees, who could ask for tighter stochastic analysis or more diagnostics on the sampling error. I would recommend peer review.

Referee Report

3 major / 2 minor

Summary. The paper introduces INEUS, a meshfree iterative neural solver for high-dimensional partial integro-differential equations (PIDEs). It reformulates the PIDE as a sequence of recursive regression problems by replacing explicit nonlocal jump integrals with single-jump Monte Carlo sampling, avoiding full residual differentiation as in PINNs. A contraction-based convergence proof is given for linear PIDEs, and numerical experiments claim accurate, scalable solutions for both linear and nonlinear high-dimensional examples.

Significance. If the single-jump sampling preserves convergence to the viscosity solution, INEUS would represent a useful advance for high-dimensional PIDEs by providing an efficient, meshfree alternative that scales better than grid-based or explicit-integral methods. The iterative regression formulation and avoidance of expensive nonlocal evaluations are practical strengths; the contraction proof for the linear case adds rigor, though its applicability to the sampled implementation is central to the contribution.

major comments (3)

[§4] §4 (Convergence analysis): The contraction mapping argument is stated for the exact integral operator of the linear PIDE. However, the algorithm in §3 approximates the nonlocal term by single-jump sampling, whose Monte Carlo variance does not vanish with iteration count and scales with dimension. No error bound or modified contraction estimate is provided that absorbs this stochastic perturbation, so the proof does not directly establish convergence of the implemented method.
[§5] §5 (Numerical experiments, nonlinear cases): No theoretical guarantee is supplied for nonlinear PIDEs, yet the abstract and experiments claim accurate solutions for nonlinear high-dimensional examples. The recursive regressions rely on the same single-jump estimator; without an analysis showing that sampling error does not prevent convergence to the viscosity solution, the experimental results remain vulnerable to post-hoc tuning of sampling or network architecture.
[§3] §3 (Method): The claim that single-jump sampling 'preserves accuracy' while enabling recursive regressions is load-bearing for both the linear proof and the nonlinear experiments. An explicit bound on the bias/variance introduced at each iteration, or a comparison against exact-integral baselines in moderate dimensions, is required to substantiate scalability claims in high dimensions.

minor comments (2)

[§2-3] Notation for the jump measure and sampling distribution should be introduced once and used consistently across the method and proof sections.
[Abstract] The abstract states 'supported by a contraction-based convergence proof,' but the proof applies only to linear PIDEs; this scope limitation should be stated explicitly in the abstract and introduction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below, providing clarifications on the scope of our results and committing to targeted revisions that acknowledge limitations while strengthening the manuscript's presentation of the method.

read point-by-point responses

Referee: [§4] The contraction mapping argument is stated for the exact integral operator of the linear PIDE. However, the algorithm in §3 approximates the nonlocal term by single-jump sampling, whose Monte Carlo variance does not vanish with iteration count and scales with dimension. No error bound or modified contraction estimate is provided that absorbs this stochastic perturbation, so the proof does not directly establish convergence of the implemented method.

Authors: We agree that the contraction mapping in §4 applies strictly to the exact integral operator. The single-jump sampling introduces persistent Monte Carlo variance that does not vanish with iterations and can scale with dimension. A full modified contraction estimate absorbing this stochastic error would require substantial additional analysis beyond the current scope. In the revision we will insert a new paragraph in §4 that explicitly discusses the approximation, notes that variance can be controlled by increasing samples per iteration, and highlights that the numerical experiments provide empirical support for convergence of the implemented algorithm despite the perturbation. revision: partial
Referee: [§5] No theoretical guarantee is supplied for nonlinear PIDEs, yet the abstract and experiments claim accurate solutions for nonlinear high-dimensional examples. The recursive regressions rely on the same single-jump estimator; without an analysis showing that sampling error does not prevent convergence to the viscosity solution, the experimental results remain vulnerable to post-hoc tuning of sampling or network architecture.

Authors: The referee is correct that our convergence analysis covers only the linear case. For nonlinear PIDEs we report only numerical accuracy. We will revise the abstract, §5, and the conclusion to state clearly that the nonlinear results are empirical observations without a supporting convergence proof. We will also add a short discussion of how sampling error is managed in practice through validation-based hyperparameter selection, thereby reducing the impression that the results rely on post-hoc tuning. revision: yes
Referee: [§3] The claim that single-jump sampling 'preserves accuracy' while enabling recursive regressions is load-bearing for both the linear proof and the nonlinear experiments. An explicit bound on the bias/variance introduced at each iteration, or a comparison against exact-integral baselines in moderate dimensions, is required to substantiate scalability claims in high dimensions.

Authors: To substantiate the accuracy claim we will add to §3 both an explicit variance expression for the single-jump Monte Carlo estimator and a new set of moderate-dimensional experiments (d=2 to d=5) that directly compare single-jump sampling against exact-integral evaluation on the same problems. These comparisons will quantify the introduced bias and variance and demonstrate that the error remains small enough to preserve the observed accuracy, thereby supporting the scalability statements for higher dimensions. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central claims rest on a contraction-mapping convergence proof for the iterative scheme applied to linear PIDEs (with the nonlocal term treated exactly) and on separate numerical experiments for both linear and nonlinear cases. No step in the provided abstract or description reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation, or ansatz that is defined by the method itself. The single-jump sampling is presented as a practical replacement for the integral operator rather than as a quantity whose accuracy is guaranteed by construction; any gap between the exact proof and the stochastic estimator is a question of error analysis, not circularity. The derivation therefore remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The method rests on the assumption that single-jump Monte Carlo sampling unbiasedly approximates the nonlocal integral and that the recursive regression sequence converges; no explicit free parameters are named in the abstract.

axioms (2)

domain assumption Single-jump sampling replaces full integral evaluation without introducing bias that prevents convergence.
Invoked when the method is introduced as replacing explicit evaluation of nonlocal jump integrals.
domain assumption The sequence of regression problems contracts to the true PIDE solution for linear cases.
Stated as supported by a contraction-based convergence proof.

pith-pipeline@v0.9.0 · 5438 in / 1242 out tokens · 32810 ms · 2026-05-08T13:06:27.787365+00:00 · methodology

discussion (0)

Reference graph

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