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Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations
Ling Guo, Hao Wu, Xiaochen Yu, Tao Zhou
TL;DR
High-dimensional FPDEs pose costly numerical challenges because their operators are nonlocal and singular. MC-PINNs use Monte Carlo estimates of fractional derivatives and an unbiased physics-loss estimate; the reported examples identify hidden parameters and solution fields, while evaluation counts compare favorably with the second-order GL scheme.
Problem
Numerical FPDE solutions can require high computational cost and memory because of nonlocality and singularity, while high-dimensional inverse fractional Laplacian problems are bottlenecks for classical methods.
Source: §1 · pp. 1, 3Method
MC-PINNs use Monte Carlo sampling to compute fractional derivatives of DNN outputs and an unbiased estimate of the physics-based loss during training.
Source: §1 · p. 2Results
MC-PINNs identified hidden FPDE parameters and solution fields, with error worsening at higher dimensions; fractional-Laplacian evaluation uses about 8 × m + 1 DNN evaluations per point versus q^d × L for second-order GL.
Source: §4.1 · p. 12, §4.2 · p. 13Takeaways & Limitations
The sampling approach alleviates computational cost and is promising for high-dimensional problems.
Source: §1 · p. 2The finite-difference-type approximation may incur rounding error and numerical instability for extremely small r.
Source: §3.2.1 · p. 7
Abstract
from arXiv · showhide
We introduce a sampling based machine learning approach, Monte Carlo physics informed neural networks (MC-PINNs), for solving forward and inverse fractional partial differential equations (FPDEs). As a generalization of physics informed neural networks (PINNs), our method relies on deep neural network surrogates in addition to a stochastic approximation strategy for computing the fractional derivatives of the DNN outputs. A key ingredient in our MC-PINNs is to construct an unbiased estimation of the physical soft constraints in the loss function. Our directly sampling approach can yield less overall computational cost compared to fPINNs proposed in \cite{pang2019fpinns} and thus provide an opportunity for solving high dimensional fractional PDEs. We validate the performance of MC-PINNs method via several examples that include high dimensional integral fractional Laplacian equations, parametric identification of time-space fractional PDEs, and fractional diffusion equation with random inputs. The results show that MC-PINNs is flexible and promising to tackle high-dimensional FPDEs.
1. Introduction
FPDE solvers face high computational and memory costs from nonlocal, singular fractional derivatives, especially in high-dimensional inverse problems. MC-PINNs combine neural-network surrogates with Monte Carlo derivative estimates and an unbiased physics-loss estimate to address these challenges.
1. Introduction
- Nonlocality and singularity make numerical FPDE solvers costly and memory-intensive, while inverse problems also require expensive forward solves to identify unknown parameters.High-dimensional inverse fractional Laplacian problems on bounded domains are identified as a bottleneck for classical numerical methods.Source: §1 · pp. 1, 3
- PINNs embed the PDE in a neural network sharing parameters with a solution surrogate, enabling training with less data while representing physical laws.The paper notes that PINNs have been applied to diverse forward and inverse problems in physics and fluid mechanics.Source: §1 · p. 2
- Fractional PINNs use numerical discretization for fractional derivatives, but the resulting computational cost grows exhaustively with physical dimension.The paper contrasts this with automatic differentiation for integer-order operators.Source: §1 · p. 2
- MC-PINNs estimate DNN-output fractional derivatives by direct Monte Carlo sampling and use an unbiased physics-based loss estimate to optimize network parameters.This replaces traditional fractional-derivative schemes such as finite differences.Source: §1 · p. 2
- The method is presented for random-parametric FPDEs that lead to uncertainty-quantification problems.The paper's examples also cover forward and inverse high-dimensional space-time fractional PDEs.Source: §1 · pp. 2–3
- The authors note that prior work on inverse space-time fractional PDEs with a fractional Laplacian seldom addressed three-dimensional problems.Classical methods for 3D space-fractional advection-diffusion equations were described as focusing mostly on the Riesz derivative, unlike the hyper-singular integral fractional Laplacian considered here.Source: §1 · p. 2
2. Notations and problem setup
The setup considers forward and inverse fractional advection-diffusion problems on a bounded spatial domain. MC-PINNs approximate the solution with a neural surrogate and optimize it against observed data and an unbiased estimate of the FPDE equation loss.
2. Notations and problem setup
- The high-dimensional fractional advection-diffusion equation is posed on bounded Ω⊂R^d, with diffusion coefficient c and mean-flow velocity v.The displayed equation includes ∂tγ, c(−△)^α/2u(x, t), and v · ∇u(x, t).Source: §2 · p. 3
- The fractional Laplacian is defined through a hypersingular integral, with P.V. denoting the principal value.The passage also introduces C_d,α as a constant.Source: §2 · p. 3
- For forward problems, coefficients and conditions are known and the solution u(x, t) is the quantity of interest; inverse problems use limited extra u-sensors to infer parameters and the full solution field.The inverse targets include α, γ, c, and v.Source: §2 · pp. 3–4
- The method trains a neural surrogate uNN(x, t) to fit observed u- and f-sensor data while optimizing an unbiased estimate of the FPDE mean-square equation loss.The authors state that this formulation reduces computational complexity relative to fPINNs and can be used for high-dimensional FPDEs.Source: §2 · p. 4
3. Methodology
PINNs represent the PDE solution with a neural-network surrogate and include the physics law in the training objective. MC-PINNs extend this approach by sampling fractional operators and constructing an unbiased estimate of the equation loss.
3.1. Physics-Informed Neural Network (PINNs)
- PINNs substitute a neural-network solution surrogate into the PDE and optimize its parameters by fitting the training data and minimizing a loss function.The surrogate takes spatial and temporal coordinates as input and can be evaluated throughout the domain after training.Source: §3.1 · pp. 4–5
3.2. Monte Carlo Physics-Informed Neural Networks (MC-PINNs)
- MC-PINNs use Monte Carlo sampling to compute fractional derivatives of the network output and estimate the physics-based equation loss without relying on traditional discretization schemes.The overall loss combines the equation loss with boundary and solution-data losses, with separate weights.Source: §3.1 · p. 5, §3.2 · p. 6
3.2.1. Stochastic approximation of fractional operators
- The spatial finite-difference-type approximation can suffer rounding error and numerical instability when the sampled distance r is extremely small.In practice, the method uses rϵ = max{ϵ, r} with a small positive ϵ.Source: §3.2.1 · p. 7
- Spatial and temporal fractional operators are approximated stochastically, while integer-order derivatives are handled by automatic differentiation.For the spatial approximation, the integral is divided into a neighborhood around the evaluation point and its complement; the time-fractional derivative is also sampled.Source: §3.2.1 · pp. 6–8
3.2.2. Unbiased estimation of the equation loss
- The training procedure samples residual points and random operator parameters, then uses the estimated equation loss in the weighted total loss to update network parameters with Adam.The estimated equation loss is unbiased when ϵ = ϵt = 0 and rounding error can be ignored.Source: §3.2.2 · pp. 8–9
4. Simulation results
The simulations assess MC-PINNs on high-dimensional fractional Laplacian equations, inverse fractional ADEs, and fractional diffusion with random inputs. Across these settings, they report low solution errors, recovery of unknown parameters, and reduced computational cost relative to fPINNs.
4.1. High-dimensional fractional Laplacian equation
- Errors for fractional Laplacian tests saturated near 10^-3 as sample number and neighborhood radius varied, while uncertainty decreased with larger settings.The tests covered fractional orders α = 0.5, 1.2, 1.5, and 1.8 and included 2D, 3D, and 10D cases.Source: §4.1 · pp. 10–11, Figure 3 · p. 10, Figure 4 · p. 11
- In multidimensional fractional-Laplacian simulations, MC-PINNs produced errors around 10^-3 and required about 8 × m + 1 evaluations versus q^d × L for second-order GL, with m=20.The authors conclude that MC-PINNs greatly alleviate computational cost compared with fPINNs and set up the potential to solve 10D problems.Source: §4.1 · pp. 11–12
4.2. Inverse problems of fractional advection-diffusion equation (ADE)
- In 1D, 3D, and 5D inverse ADE problems, inferred parameters converged to their true values in fewer than 20,000 epochs, and the solution fields were well identified.The reported solution error worsened as physical dimension increased; Table 1 summarizes identified parameters and relative errors for the synthetic-data problems.Source: §4.2 · p. 13, Figure 6 · p. 13, Figure 7 · p. 13
4.3. Fractional diffusion equation with random inputs
- For random-input fractional diffusion, MC-PINNs relative L2 error increased for higher-dimensional problems when the same number of residual points was used during training, with α0=1, α=α0, and µ=0.The 5D plots show the recovered solution and absolute error compared with the fabricated solution.Source: §4.3 · p. 14, Figure 8 · p. 15
- With five sensors in a 2D diffusion problem, approximate Bayesian computation was used to estimate posterior distributions for the unknown parameters α and µ.The posterior used 10^5 prior draws and tolerance 2.5 × 10^-4; Figure 9 marks the true parameter values.Source: §4.3 · p. 14, Figure 9 · p. 15
5. Summary
MC-PINNs estimates FPDE constraints in the loss function without the substantial auxiliary-point growth associated with discretizing fractional derivatives. The paper demonstrates the method on high-dimensional integral fractional Laplacians, time-space fractional parameter identification, and fractional diffusion with random inputs.
5. Summary
- MC-PINNs uses Monte Carlo quadrature to estimate the FPDE constraint in the loss function, reducing the growth in auxiliary points for high-dimensional problems.The paper identifies extensions to more general nonlocal problems and peridynamic models as future work.Source: §5 · p. 15
- The method is demonstrated for high-dimensional integral fractional Laplacians, time-space fractional parameter identification, and fractional diffusion with random inputs.Source: §5 · p. 15
- The paper proposes future extensions to more general nonlocal problems and peridynamic models.Source: §5 · p. 15