nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications

Guofei Pang; Marta D'Elia; Michael Parks; George E. Karniadakis

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nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications

Guofei Pang, Marta D'Elia, Michael Parks, George E. Karniadakis

arXiv:2004.04276v1Apr 8, 2020math.APcs.LGmath.OCphysics.comp-ph

TL;DR

The paper addresses how to identify nonlocal model parameters and functions from limited, noisy data when parameter changes can alter the operator itself. It extends PINNs into nPINNs, introducing a universal operator and a variable-order turbulence model. The results show accurate forward solutions, operator mimicking among parameter estimates, and universal parameter behavior in turbulent Couette flow.

Problem
Nonlocal parameters and kernels can be unknown, noisy, sparse, and difficult to identify because parameter changes may substantially alter the operator and its spectral properties.
Method
The paper combines neural-network surrogates with governing equations and a unified nonlocal operator to infer parameters and variable-order functions without deriving adjoint equations.
Results
nPINNs are as accurate as other discretization methods for forward problems, while also estimating nonlocal parameters and variable orders in Poisson and turbulent Couette-flow applications.
Takeaways & Limitations
The inferred δ and α(y+) exhibit universal behavior with Reynolds number, and finite interaction radii can provide computational savings compared with δ = ∞.
Takeaways & Limitations
Identifying the kernel function γ remains an open, substantially more complex problem because it requires learning a functional form rather than model parameters.

Abstract

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Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integral equations with sparse, noisy, unstructured, and multi-fidelity data. PINNs incorporate all available information into a loss function, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal operator, which converges to the classical Laplacian as one of the operator parameters, the nonlocal interaction radius $δ$ goes to zero, and to the fractional Laplacian as $δ$ goes to infinity. This universal operator forms a super-set of classical Laplacian and fractional Laplacian operators and, thus, has the potential to fit a broad spectrum of data sets. We provide theoretical convergence rates with respect to $δ$ and verify them via numerical experiments. Second, we use nPINNs to estimate the two parameters, $δ$ and $α$. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon: different pairs of estimated parameters could produce multiple solutions of comparable accuracy. Third, we propose another nonlocal operator with spatially variable order $α(y)$, which is more suitable for modeling turbulent Couette flow. Our results show that nPINNs can jointly infer this function as well as $δ$. Also, these parameters exhibit a universal behavior with respect to the Reynolds number, a finding that contributes to our understanding of nonlocal interactions in wall-bounded turbulence.

1 Introduction

The paper develops nPINNs for discovering nonlocal models from limited, noisy measurements, combining neural-network surrogates with a unified nonlocal operator. It introduces a parameterized operator spanning classical and fractional Laplacians, identifies operator mimicking, and applies variable-order modeling to turbulent Couette flow.

Motivation: Nonlocal models represent long-range interactions and can describe multiscale and anomalous-diffusion phenomena that standard PDEs may miss.Examples include fracture, failure, super-diffusion, and sub-diffusion.
Motivation: Identifying nonlocal parameters from limited measurements is difficult because changing parameters can substantially alter the operator itself and its spectral properties.The parameters may also be non-measurable, sparse, and noisy.
Approach: The proposed framework combines machine learning, physical principles, universal operators, and neural-network surrogates without requiring adjoint-equation derivations.Governing equations act as implicit regularization in the objective, potentially reducing training-data requirements.
Contributions: The universal operator is parameterized by interaction radius δ and decay rate α, reducing to the classical Laplacian as δ → 0 and converging to the fractional Laplacian as δ → ∞.The paper provides theoretical convergence rates in δ and verifies them numerically.
Contributions: nPINNs estimate δ and α, but strong loss-function non-convexity can produce operator mimicking: distinct parameter pairs generate operators equally effective on training data.These solutions correspond to multiple good local minima with comparable accuracy.
Contributions: For turbulent Couette flow, nPINNs jointly infer δ and a spatially variable decay exponent α(y+), finding universal behavior with respect to Reynolds number.The paper presents this as a contribution to understanding nonlocal interactions in wall-bounded turbulence.

2 A nonlocal Poisson problem

The paper formulates nonlocal Poisson problems through integral operators whose kernels encode long-range interactions and whose parameters determine solution regularity. Finite interaction neighborhoods require nonlocal boundary conditions on an interaction domain.

Nonlocal operator: A nonlocal Laplace operator acts on a scalar function through an integral involving a nonnegative symmetric kernel γ(x,y;µ) and parameters µ.The integral formulation supports long-range interactions and relaxes differentiability requirements on solutions.
Nonlocal operator: The kernel parameters µ are application-dependent, usually unknown and non-measurable, and determine regularity properties of the solution.Their identification is the focus of the paper.
Interaction domain: For a bounded domain Ω, the interaction domain ΩI contains exterior points that interact with points inside Ω through a nonzero kernel.Conditions on this interaction domain are needed for existence and uniqueness of the nonlocal Poisson problem.
Interaction domain: With finite interaction length, the kernel acts only within Bδ(x), the Euclidean ball of radius δ centered at x, making δ the interaction radius.The resulting interaction domain is denoted Ωδ and consists of exterior points within distance δ of points in Ω.
Interaction domain: Figure 1 depicts the physical domain Ω, its interaction domain Ωδ, and the kernel-support neighborhood Bδ(x).These regions visualize where the nonlocal operator couples points.
Nonlocal Poisson problem: The nonlocal Poisson problem seeks u on Ω∪Ωδ from source data f in Ω and volume data g on Ωδ.The condition on Ωδ is the nonlocal analogue of a Dirichlet boundary condition, called a Dirichlet volume constraint.

3 A unified nonlocal operator and its relationship with classical and fractional Laplacian

The paper introduces a unified nonlocal Laplace operator parameterized by interaction radius δ and decay rate α, connecting classical and fractional Laplacians through limiting behavior. It establishes convergence rates and positions the operator as a flexible model for nonlocal phenomena.

The unified operator is parameterized by interaction radius δ and kernel decay rate α, with α ∈ (0, 2).
The paper formulates joint estimation of δ and α from limited observations of u in Ω∪Ωδ and f in Ω.
As δ → 0, the properly scaled operator converges to the classical Laplacian, while as δ → ∞ it converges to the fractional Laplacian.
The operator’s large-δ approximation to the fractional Laplacian has error scaling as δ^max{α−2,−α}.
The operator forms a superset of the classical and fractional Laplacians and is intended to fit a broad spectrum of experimental data.
The unified operator excludes tempered fractional operators, although multiplying the kernel by exp{−λ∥x−y∥} can extend nPINNs to generalized tempered operators.

4 Nonlocal Physics-Informed Neural Networks (nPINNs)

nPINNs approximate the solution with a fully connected neural network and jointly optimize network parameters with unknown operator parameters through a loss combining equation residuals and data mismatch. The framework supports forward and inverse nonlocal problems, while strong non-convexity can produce multiple comparably accurate parameter pairs.

4 Nonlocal Physics-Informed Neural Networks (nPINNs): The algorithm collects solution observations, evaluates residuals and misfit terms, and minimizes the loss with respect to neural-network and operator parameters.
4 Nonlocal Physics-Informed Neural Networks (nPINNs): The loss combines a physics-driven residual term with a data-driven mismatch term, with the residual acting as a regularizer that improves minimization conditioning.
4 Nonlocal Physics-Informed Neural Networks (nPINNs): nPINNs use a fully connected neural network to approximate the solution and optimize its parameters together with unknown nonlocal parameters.
4.1 Fully-connected neural networks: Neural-network depth and width are selected heuristically, and optimization convergence can be sensitive to these hyper-parameters.
4.2 Evaluation and minimization of the loss function: For inverse problems, observations inside Ω improve learning, whereas forward problems can require only interaction-domain data when the nonlocal volume constraint is known.
4.2 Evaluation and minimization of the loss function: The loss is strongly non-convex, so multiple local minima may yield distinct operators with comparable loss values and similarly accurate solutions.

5 Computational results

Computational experiments test the unified operator’s classical and fractional limits, convergence rates, forward-solution accuracy, and nPINN parameter-estimation behavior. Results support the operator’s limiting properties and show that accuracy and optimization outcomes depend on discretization, observations, parameters, and initialization.

5.1 Limit behavior of the parametrized operator: For small δ, the unified operator reduces to the classical Laplacian regardless of α; for large δ, it reduces to the fractional Laplacian.These limits were confirmed numerically for the nonlocal Poisson problem.
5.1 Limit behavior of the parametrized operator: With fixed α = 0.5, solutions span the spectrum between classical and fractional limits as δ varies from zero to infinity.Changing α further broadens the range of solutions represented by the operator.
5.1 Limit behavior of the parametrized operator: The relative-error rates are O(δ^(2−α)) as δ →0 and O(δ^max{α−2,−α}) as δ →∞, matching theoretical predictions.The numerical convergence tests confirmed the expected rates for the classical and fractional limits.
5.2 Accuracy of nPINNs for forward problems: nPINNs and fPINNs show similar convergence with residual-point count, while small-N behavior matches the finite difference scheme.Increasing N improves residual evaluation but can make optimization harder after a threshold, causing error stagnation.
5.2 Accuracy of nPINNs for forward problems: nPINNs accurately matched reference solutions, achieving ϵ=1.7e-03 for problem (I), ϵ=4.9e-04 for problem (II), ϵ=1.7e-04 in two dimensions, and ϵ=1.1e-03 in three dimensions.The reported tests used 600 residual points for the one-dimensional problems, 300 in two dimensions, and 400 in three dimensions.
5.3 Parameter estimation for determining the operator: Parameter-estimation outcomes depend on true parameters and initialization because the loss is non-convex, producing global or comparable local minima and, sometimes, higher-loss solutions.Observation count also interacts with parameters and initialization: increasing observations can improve estimates, but optimization errors mean more observations are not always better.

6 Application to turbulence modeling of Couette flow

The paper applies nPINNs to turbulent Couette flow using a variable-order nonlocal operator for total shear stress, jointly inferring the horizon δ and profile α(y+). Results show Reynolds-number-universal profiles, accurate stress prediction, and multiple mimicking operators influenced by δ initialization.

Nonlocal Couette model: The Couette-flow model replaces the local RANS shear-stress equation with a variable-order nonlocal operator parameterized by δ and α(y+).The fractional order varies spatially in wall coordinates and is constrained to (0,1).
Operator limits: At infinite interaction radius, the variable-order operator reduces to a combination of left- and right-sided variable-order Caputo fractional derivatives.Classical local behavior is recovered only under additional limiting conditions and, for this Couette model, only in the viscous sublayer.
nPINN parameterization: nPINNs represent α(y+) with a fully connected neural network while separately approximating the velocity profile U+(y+).The method jointly optimizes the neural-network parameters for the solution and fractional-order profile.
Simulation results: For fixed δ initialization, learned α(y+) profiles nearly coincide across Reτ=125, 180, and 250, indicating Reynolds-number-universal fractional-order behavior.The profile is nearly one near the walls and decreases toward the centerline as nonlocal effects intensify.
Simulation results: The two initial guesses δ0=100 and δ0=1e10 produce distinct operators whose actions on the reduced RANS equation are the same, with overlapping computed stress profiles.Figure 18 compares the resulting Reynolds stresses with DNS data for both initial guesses and different Reynolds numbers.
Sensitivity to initialization: For δ0∈{1, 10, 100, 1000, 1e10}, larger initial horizons reduce final loss, while δ≥100 yields loss values of the same order and can produce mimicking operators.Using Δτ+<0.01, only the profiles associated with δ0=1000 and 1e10 are acceptable; slightly relaxed thresholds also accept δ0=100.

7 Summary

The paper presents a unified nonlocal operator and nPINN framework for forward and inverse problems, spanning classical and fractional Laplacians. It reports accurate forward solutions, Reynolds-number-linked turbulence parameters, and open directions for kernel identification and tempered operators.

The unified operator spans broad nonlocal elliptic operators by tuning α and δ, recovering classical and fractional Laplacians in specified limits.It is equivalent to the classical Laplacian as δ →0 for α ∈(0, 2) or α →2 with δ = ∞, and to the fractional Laplacian as δ = ∞ with α ∈(0, 2).
The algorithm treats forward and inverse problems similarly, optimizing neural-network parameters alone or jointly with model parameters, while allowing black-box operator discretizations.
nPINNs match other discretization methods for forward problems, with optimal error convergence in residual-point count, rough-solution support, and applicability in any dimension.
Turbulence experiments show nPINNs can learn both the solution and variable order in high-dimensional parameter spaces, with α(y+) and δ exhibiting universal Reynolds-number behavior.
Identifying the kernel function γ remains an open, higher-dimensional problem beyond parameter identification, while variable decay-rate learning provides an initial step.
Tempered fractional operators are outside the operator family Lδ,α, although extending the framework to generalized tempered operators is described as straightforward ongoing work.

Appendix A Proof of Lemma 3.1

The appendix proves convergence properties of the unified operator in two and three dimensions using coordinate transformations and Taylor expansions. The large-radius limit follows from decay of the difference from the fractional Laplacian.

The two-dimensional proof transforms the operator into polar coordinates and assumes u ≡0 outside Ω before expanding the integrand.
The first Taylor-expansion integral vanishes by angular symmetry, while the second produces a term proportional to δ^(2−α)Δu.
For δ →0, the remaining integral is of order δ^β with β > 2 − α, so the third Taylor term can be neglected.
The large-δ proof compares the fractional Laplacian with the unified operator and establishes an asymptotic convergence rate δ^(α−2).

B.1 The one-dimensional case

The one-dimensional implementation evaluates the nonlocal operator at residual points by separating singular and nonsingular regions according to δ. Taylor expansion handles near-singular terms, while composite quadrature evaluates the remaining integrals.

At residual points xk ∈Ω, the method prescribes nonlocal boundary values and uses the neural-network solution inside Ω.
Residual points are excluded from boundary layers and neighborhoods of jump discontinuities to avoid integrating near discontinuities.
For δ ≤ρ, the integral is evaluated inside Ω using the Taylor expansion.
For ρ < δ ≤D, the integral is split around the singularity, with near-field terms approximated by Taylor expansion and outer terms evaluated by composite Gauss quadrature.
For δ >D, Taylor expansion handles the first sub-integral, composite Gauss quadrature handles the second and third, and volume constraints rewrite the remaining terms.
The cases are unified using a0 = min{δ, ρ}, am = min{δ, D}, and a Heaviside function H.

B.2 The multivariate case

The multivariate implementation extends the one-dimensional quadrature strategy to polar and spherical coordinates. Annular and spherical-shell domains are partitioned into segments, each integrated with Gauss–Legendre quadrature.

The multivariate construction extends the preceding one-dimensional operator-evaluation procedure to two- and three-dimensional settings.
Two-dimensional case: In two dimensions, the integration annulus uses polar coordinates with radial bounds a0 = min{δ, ρ} and am1 = min{δ, D}.
Two-dimensional case: The two-dimensional annulus is divided into radial-angular segments, with Gauss–Legendre quadrature points and weights assigned within each segment.
Three-dimensional case: In three dimensions, the integration region is a spherical shell parameterized by radial distance, azimuth, and polar angle.
Three-dimensional case: The three-dimensional shell is partitioned into m1 × m2 × m3 segments over the radial and angular coordinates for Gauss–Legendre integration.

Appendix C Quadrature accuracy

The appendix evaluates composite Gauss quadrature accuracy for one-, two-, and three-dimensional tests using fixed operator parameters and prescribed test-point sets. Figure 20 examines how sub-integral counts and Gauss–Legendre points affect relative error and total quadrature cost.

The quadrature tests use decay exponent α = 1.5 and interaction radius δ = 10^100.
Test points are generated from Sobol sequences while excluding a neighborhood of the domain boundary.The domain diameter is D = 2 and the exclusion parameter is ρ = 10^-5.
Figure 20 varies the number of sub-integrals m and Gauss–Legendre points per sub-integral M for composite quadrature.The figure reports relative error on the left axis and total quadrature points on the right axis.
The figure states that quadrature accuracy drops as m and M increase, while higher accuracy requires more total quadrature points.The total point counts are mM, m^2M^2, and m^3M^3 in one, two, and three dimensions, respectively.