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Convergence of kernel and neural-network methods for Hamilton--Jacobi--Bellman equations on unbounded domains

Yumiharu Nakano

arXiv:2610.09299v1math.NAcs.NA

TL;DR

Empirical squared residuals for non-monotone, mesh-free HJB approximations do not directly supply the pointwise control needed for viscosity-solution error estimates. The paper combines empirical-to-population control, interpolation, and a controlled-diffusion truncation estimate for kernel collocation and neural networks. It establishes compact-set convergence and quantitative error bounds under the methods’ respective regularity assumptions.

  • Problem

    Empirical squared residuals do not directly provide the pointwise residual control required for viscosity-solution error estimates.

    Source: §1 · p. 2

  • Method

    The paper turns empirical residual losses into pointwise control using norm constraints, interpolation, and a controlled-diffusion estimate for truncation error.

    Source: §1 · p. 2, §2.3 · p. 6

  • Results

    Kernel collocation has compact-uniform convergence and an a posteriori quantitative error bound, while neural networks have compact-uniform convergence in probability and a high-probability quantitative bound.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The resulting kernel bounds are governed by fill distance, while neural-network bounds depend on network and sample sizes.

    Source: §1 · pp. 2–3

    The reported training runs use a penalty that differs from the Sobolev constraint, so the neural-network convergence theorems do not apply directly to those runs.

    Source: §5.3 · p. 27

Abstract

from arXiv · show

We derive error bounds from empirical squared residuals for two non-monotone, mesh-free approximations of Hamilton--Jacobi--Bellman equations on unbounded spatial domains: collocation with Wendland kernels under a native space norm constraint, and physics-informed neural networks with smooth activations under a Sobolev-norm constraint. The main step is to recover the pointwise residual control needed for viscosity-solution error estimates from the empirical loss, while quantifying the discretization or sampling error. For the kernel method, we prove uniform convergence on compact sets under classical and native space regularity and derive an a posteriori quantitative error bound for the unique bounded continuous viscosity solution. For the neural-network method, we prove uniform convergence on compact sets in probability and give a quantitative high-probability error bound under classical Sobolev regularity. The bounds are governed by the fill distance for kernels and by the network and sample sizes for neural networks. The proofs combine empirical-to-population estimates, an interpolation inequality that turns an $L^2$ residual into a sup norm bound, and a controlled diffusion estimate for the error caused by truncating the domain. The regularity assumptions used to obtain convergence from approximation properties are additional to viscosity wellposedness. Numerical experiments in one and two space dimensions illustrate the methods and the estimates.

1 Introduction

The paper develops empirical-residual error bounds for non-monotone, mesh-free approximations of finite-horizon second-order HJB equations on unbounded domains. It bridges empirical L2 losses to the pointwise residual control required for viscosity-solution estimates, then applies the framework to kernel collocation and neural networks.

Motivation and contribution

  • The central gap is that empirical squared PDE residuals and terminal mismatch do not provide the pointwise control required by viscosity-solution error estimates.The analysis recovers pointwise residual control from computable empirical L2 losses while quantifying kernel discretization and neural-network sampling error.Source: §1 · p. 2
  • The shared mechanism combines norm constraints for locally uniform C3-control, L2-to-sup-norm interpolation, and a controlled-diffusion tail estimate for domain truncation.Together these steps convert residual bounds on truncation cylinders into convergence on compact subsets of the unbounded domain.Source: §1 · p. 3

Kernel realization

  • Wendland-kernel collocation under a native-space constraint yields compact-uniform convergence under additional classical and native-space regularity, plus an a posteriori error bound under (A1) and (A2).The error bound depends on empirical loss and fill distance and applies to the unique bounded continuous viscosity solution.Source: §1 · p. 3

Neural-network realization

  • The neural-network realization uses smooth activations and an Hs-norm constraint, with qualitative and quantitative bounds governed by network approximation and sampling complexity.The convergence rate uses a Rademacher bound obtained from a covering argument in network parameters.Source: §1 · p. 3

Assumptions and scope

  • Convergence from approximation properties requires separate classical and native-space or Sobolev regularity assumptions beyond viscosity wellposedness.Fully nonlinear equations without a stochastic control representation are outside the stated scope.Source: §1 · p. 3

2 Setting and two basic estimates

The section sets the HJB problem on expanding truncation cylinders and establishes two shared steps: converting L2 residual control to a sup-norm bound, and converting cylinder error to compact-domain approximation error.

2.1 Problem setting and truncation domains

  • Under (A1) and (A2), the HJB equation has a unique bounded continuous viscosity solution, while convergence from approximation properties additionally assumes classical and stated function-space regularity.The Sobolev regularity is a separate assumption because it also requires smoothness and decay at infinity.Source: §2.1 · p. 5

2.2 From L2 to sup-norm bounds

  • On bounded boxes, Hölder control lets the interpolation estimate convert an L2 bound into a sup-norm bound, with constants uniform in the truncation index for the stated cylinders and cubes.For the residual, the paper uses α=1 and dimension d+1; higher-order residual regularity is generally unavailable because the Hamiltonian is only Lipschitz in gradient and Hessian.Source: §2.2 · pp. 5–6

2.3 From residual bounds to approximation error

  • Theorem 2.4 bounds compact-set solution error using residual and terminal errors on truncation cylinders, with a controlled-diffusion tail estimate accounting for the unbounded domain.The stochastic representation is used to control the error outside the truncated domain.Source: §2.3 · pp. 6–7

3 Kernel realization

The kernel realization uses constrained Wendland-kernel collocation to turn empirical residual losses into compact-set convergence and a posteriori error control, with fill distance and truncation entering the bounds.

3.1 Kernel ansatz and native space

  • Wendland-kernel translates on a uniform space-time grid form the candidate, while a native-space norm constraint supplies locally uniform C3 control.The native space is equivalent to a Sobolev space under the stated Fourier-decay condition.Source: §3 · p. 8, §3.1 · pp. 8–9

3.2 Empirical loss and a posteriori bound

  • A fill-distance quadrature estimate controls the difference between empirical grid losses and population losses for Lipschitz residual and terminal-mismatch functions.The stated quadrature error scales with the rescaled fill distance R_n h_n.Source: §3.2 · p. 9

3.3 Wendland kernel interpolation rate

  • The Wendland interpolation estimate is uniform over the expanding cylinder family when R_n h_n is sufficiently small.It supplies the interpolation step used to control candidate residuals in the qualitative convergence argument.Source: §3.3 · p. 10

3.4 Qualitative convergence

  • Under classical and native-space regularity, kernel candidates satisfying the stated near-optimality condition converge uniformly on compact sets.The comparison condition uses the exact solution's kernel interpolant and is satisfied by every feasible ε-optimal solution.Source: §3.4 · pp. 10–11

3.5 Quantitative error bound

  • Theorem 3.4 gives an a posteriori compact-set error bound for the unique bounded continuous viscosity solution, governed by empirical losses, fill distance, native-space radius, and truncation radius.The certificate applies to every candidate satisfying the native-space constraint.Source: §3.5 · pp. 11–12

4 Application to physics-informed neural networks

This section develops Sobolev-constrained physics-informed neural networks with smooth bounded activations, using local Sobolev control and network-parameter conditions to establish approximation and generalization tools. It proves compact-uniform convergence in probability and a high-probability quantitative error bound, with an explicit rate up to logarithmic factors under an approximation-rate assumption.

4 Application to physics-informed neural networks

  • The analysis assumes nonconstant, smooth activations with bounded derivatives, including tanh, and excludes ReLU and unbounded smooth activations.For unbounded smooth activations, network outputs may grow at infinity, preventing the global bound required by Theorem 2.4.Source: §4 · p. 12

4.1 Network class and training problem

  • The network class fixes depth while allowing width and weight bounds to grow, and constrains candidate networks by a Sobolev norm on the truncated domain.Under the activation condition, the networks have a global C1,2 bound of order (W_n B_n)^(2L+1).Source: §4.1 · pp. 13–14

4.2 Approximation in Sobolev norm

  • On compact sets, networks can approximate a function in Sobolev norm while keeping their Sobolev norms bounded by the target norm plus the approximation tolerance.Proposition 4.3 gives no control over the required widths or weight bounds, so Theorem 4.9 imposes separate generalization and tail conditions.Source: §4.2 · pp. 14–15

4.3 Generalization on the network class

  • The generalization analysis converts empirical L2 residual control to population L2 control on a Sobolev-constrained network class.Its bound is polynomial in network width and logarithmic in weight bounds and domain radius; fixed depth enters only through the constant.Source: §4.3 · pp. 16–18

4.4 Qualitative convergence

  • Theorem 4.4 establishes local uniform convergence in probability for Sobolev-constrained network approximations.The supplied passage states the theorem’s conclusion but does not provide its assumptions or rate.Source: §4.4 · p. 18
  • Under the stated assumptions and Sobolev regularity, ε_n-optimal networks converge uniformly on every compact space-time set in probability to the classical solution.The classical solution coincides with the unique bounded continuous viscosity solution under (A1) and (A2).Source: §4.4 · pp. 18–19

4.5 Quantitative error bound

  • For every compact Q ⊂ [0,T] × R^d, constants C_Q > 0 and n_0 independent of n and ρ exist; for n ≥ n_0, ρ ∈ (0,1), |v(t,x)−v_n(t,x)| ≤ C_Q with probability at least 1−ρ.Source: §4.5 · p. 20
  • The quantitative bound separates approximation, generalization, concentration, and optimization errors; domain-volume factors, Sobolev-radius factors, and truncation each contribute distinct weakening terms.The passage attributes powers of R_n to truncated-domain volumes, powers of 1 + λ_n to L2-to-sup-norm conversion, and an exponential term to truncation of the unbounded state space.Source: §4.5 · p. 20
  • With approximation error δ_n = O(M_n^-κ) and the stated parameter choices, the compact-set error is O_P(M_n^-β) up to logarithmic factors, where β = κ/((2κ + 1)(d + 3)).For tanh networks with L = 2 and v in W^{s+k,∞}, the passage gives κ = k/(d + 1).Source: §4.5 · p. 20

5 Numerical experiments

The experiments use a manufactured, spatially localized HJB problem in one and two dimensions to examine refinement, sampling, regularization, and domain truncation. Kernel errors generally decrease with refinement, while neural-network accuracy improves with width but not systematically with sample size at the tested training budget.

5.1 Test problem

  • The test uses a smooth, spatially localized version of Guo et al.'s benchmark, with active diffusion and drift switching inside the core and non-negligible values at the artificial boundary.The test solution satisfies the stated regularity hypotheses for Sections 3 and 4.Source: §5.1 · p. 21

5.2 Kernel realization: implementation and results

  • Core RMS error falls below 10^-3 on the two finest one-dimensional kernel grids, while γn reaches its minimum at Nx=17 and then rises, indicating an optimization floor.The refinement study uses R=3 and c=1.Source: §5.2 · p. 22, Figure 5.1 · p. 23
  • The one-dimensional bandwidth fit fails at c=0.5 but is relatively insensitive to c≥1; c=2 largely removes the boundary error layer seen at c=1.Optimization appears to limit performance for c≥1, and the boundary layer distinguishes core from full-cylinder errors.Source: §5.2 · p. 22, Figure 5.2 · p. 24
  • Under theorem-scaled refinement, core RMS error decreases from 4.1 × 10^-1 to 7.1 × 10^-4, although the experiment does not verify the strict norm or asymptotic optimization conditions.All computed native-space norms remain below λ, and the finest-grid empirical loss and γn rise slightly.Source: §5.2 · p. 25, Table 5.3 · p. 25
  • In two dimensions, all errors and γn decrease monotonically; the two finest γn values roughly follow the h_x^3/2 reference, while the fitted core rate remains pre-asymptotic.The finest dense-kernel solve takes 269 seconds.Source: §5.2 · p. 25, Figure 5.3 · p. 26
  • At the finest finite-difference setting, core RMS is 6.4 × 10^-4 for the kernel method versus 2.8 × 10^-3 for the monotone finite-difference baseline.The finite-difference method is much cheaper; its reported runtime is about 0.1 s versus 71 s for the kernel method.Source: §5.2 · p. 25, Table 5.5 · p. 26

5.3 Neural-network realization: architecture and Sobolev-regularized loss

  • The three-hidden-layer tanh networks use a layer-norm penalty only as an experimental proxy; it neither bounds higher derivatives nor enforces the theoretical Sobolev constraint.Consequently, the convergence theorems do not apply directly to these training runs.Source: §5.3 · p. 27

5.4 Neural-network realization: numerical results

  • For the tested networks, error decreases with width, sample size has little systematic effect, and increasing the penalty worsens both errors; the kernel has slightly lower core RMS while the network has slightly lower core maximum error.The comparison favors the networks through oracle iterate selection; the experiments illustrate rather than validate the theory, using one manufactured equation in d≤2 and short training budgets.Source: §5.4 · p. 28, Table 5.6 · p. 28

Code availability

The code, numerical data, and figure- and table-generation scripts are publicly available. The author also discloses using Codex for code refactoring, plotting code, and prose editing, while taking responsibility for the paper’s content.

Code availability

  • The code, numerical data, and scripts used to generate Section 5 figures and tables are publicly available.https://github.com/yumiharu-nakano/hjb-residual-methodsSource: Code availability · p. 29

AI disclosure statement

  • Codex was used to refactor and document Python code, generate routine plotting code, and edit author-written prose.The author states that all AI-assisted material was critically reviewed and tested, and the mathematical arguments, experimental design, numerical results, and conclusions were independently verified.Source: AI disclosure statement · p. 29
  • The author takes full responsibility for the paper’s content.Source: AI disclosure statement · p. 29
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