Source-linked AI summary
Distributed physics informed neural network for data-efficient solution to partial differential equations
Vikas Dwivedi, Nishant Parashar, Balaji Srinivasan
TL;DR
Practical PDEs often lack known analytical solutions, while mesh-based methods face difficulties with complex geometries and discretization errors. The paper proposes DPINN, which partitions the domain into cells with local PINNs and interface constraints; it reports improved data efficiency and accuracy on reported tests, including a low-Reynolds-number steady Navier–Stokes solution.
Problem
Practical PDEs often lack known analytical solutions, while mesh-based methods can struggle with complex-geometry meshing and discretization errors.
Source: §1 · pp. 1–2Method
DPINN partitions the computational domain into cells, places a local PINN in each, and adds interface constraints to the loss.
Source: §1 · p. 2, §3.1 · p. 6, §5 · p. 14Results
DPINN is reported as more data-efficient, represents and advects a high-frequency wave packet where PINN fails, improves Burgers’ equation accuracy with less data, and solves steady Navier–Stokes at low Reynolds number.
Source: §5 · p. 14Takeaways & Limitations
The authors present domain partitioning and interface constraints as a way to break a difficult problem into smaller subproblems and boost network representation capability.
Source: §3.1 · p. 6, §5 · p. 14The demonstrated Navier–Stokes case is low-Reynolds-number and steady; unsteady full three-dimensional cases require further algorithmic advances and greater computational resources.
Source: §5 · p. 14, §4.3 · p. 14
Abstract
from arXiv · showhide
The physics informed neural network (PINN) is evolving as a viable method to solve partial differential equations. In the recent past PINNs have been successfully tested and validated to find solutions to both linear and non-linear partial differential equations (PDEs). However, the literature lacks detailed investigation of PINNs in terms of their representation capability. In this work, we first test the original PINN method in terms of its capability to represent a complicated function. Further, to address the shortcomings of the PINN architecture, we propose a novel distributed PINN, named DPINN. We first perform a direct comparison of the proposed DPINN approach against PINN to solve a non-linear PDE (Burgers' equation). We show that DPINN not only yields a more accurate solution to the Burgers' equation, but it is found to be more data-efficient as well. At last, we employ our novel DPINN to two-dimensional steady-state Navier-Stokes equation, which is a system of non-linear PDEs. To the best of the authors' knowledge, this is the first such attempt to directly solve the Navier-Stokes equation using a physics informed neural network.
1. Introduction
PDEs are important across physics, engineering, and finance, but practical problems often lack known analytical solutions and mesh-based numerical methods face difficulties with complex geometries and truncation error. The paper proposes DPINN to address limitations of conventional PINNs through domain partitioning and local networks.
- Complex geometries make mesh generation difficult, while PDE discretization can introduce serious truncation errors.Source: §1 · pp. 1–2
- Prior neural-network approaches have addressed stationary PDEs with complex geometries, high-dimensional PDEs, and initial-boundary-value problems.The section cites Berg & Nyström (2018), Sirignano & Spiliopoulos (2018), and earlier work by Lagaris et al. (1998, 2000).Source: §1 · p. 2
- PINNs face vanishing-gradient concerns as PDE order increases, difficulty representing sharp local gradients over broad domains, and uncertainty about data and network-depth requirements.Source: §1 · p. 2
- DPINN partitions the domain into cells with simpler local PINNs, adding interface-flux mismatch penalties while simplifying the local function or PDE representation.The authors state this makes DPINN more data-efficient than the original PINN.Source: §1 · pp. 2–3
2. Review of physics informed neural network
The reviewed PINN approach embeds prior physical information in neural-network learning through a cost function that penalizes nonphysical outputs. Its mathematical formulation approximates the solution with a network and minimizes losses associated with the PDE and its initial and boundary conditions.
2. Review of physics informed neural network
- PINN incorporates prior physics into learning through a cost function that penalizes nonphysical solutions, supporting generalization in small-data settings.Source: §2 · p. 3
2.0.1. Mathematical formulation
- The PDE solution u is approximated by the neural-network output f, with network weights and biases defining its layers.The displayed example uses a two-layer network with activation function φ.Source: §2.0.1 · p. 3
- Automatic differentiation computes the PDE and time-derivative expressions, and gradient-based optimization minimizes the total loss.Source: §2.0.1 · pp. 4–5
- The PINN loss combines PDE-residual, boundary-condition, and initial-condition losses, whose errors are computed from network predictions and problem data.Source: §2.0.1 · p. 4
3. Distributed PINN
DPINN divides a computational domain into cells, places a separate PINN in each, and adds interface constraints to conventional PINN losses. The formulation combines cell-level PDE, boundary, and initial-condition losses with interface continuity and differentiability losses.
3. Distributed PINN
- DPINN partitions the domain into cells and uses local PINNs, with continuity and differentiability constraints acting as local regularizers.The approach is motivated by finite-volume methods; interface conditions depend on the differential operator.Source: §3 · p. 5, §3.1 · p. 6
3.1. Mathematical formulation
- For the 1D unsteady problem, the space-time domain is divided into rectangular cells, each with its own network, weights, biases, and collocation points.Source: §3.1 · p. 6
3.1.1. Expressions for the conventional PINN losses
- Each cell's conventional PINN losses are combined with additional interface losses in the total objective.The cited formulation identifies the combination but does not supply the individual conventional-loss expressions in these passages.Source: §3.1 · p. 6
3.1.2. Expressions for the additional interface losses
- Interface losses sum continuity and differentiability penalties; the implementation computes cell losses, calculates interface losses, sums them, and minimizes the total.The formulation is stated to extend to higher-dimensional problems without special adjustment.Source: §3.1.2 · pp. 7–8
4. Results and Discussions
The evaluation tests PINN representation of complex profiles, then assesses DPINN on advection, Burgers’ equation, and a two-dimensional steady Navier–Stokes problem. DPINN improves the reported Burgers’ error with fewer collocation points, and its lid-driven cavity predictions agree with established centerline results.
4.1. Advection equation
- The original PINN failed to represent the complex advection initial profile using 25,600 randomly positioned collocation points.The authors attribute the difficulty to sharp local gradients, which global regularization struggles to capture; a deeper network may slow training through vanishing gradients.Source: §4.1 · p. 9, Figure 4 · p. 10
- DPINN models the advection profile with simpler cellwise networks and interface conditions; the authors attribute its robustness to this piecewise approximation.The domain used 25 cells along x and five along t, with two-layer, five-neuron networks in each cell.Source: §4.1 · pp. 10–11, Figure 5 · p. 10
4.2. Burgers’ equation
- 2.6e-04 relative L2-norm error for DPINN versus 4.9e-04 for PINN, with about half the collocation data, on Burgers’ equation.DPINN used 4,141 collocation points; the cited PINN result used approximately 10,000.Source: §4.2 · pp. 11–12, Figure 6 · p. 11
4.3. Navier-Stokes equation
- For the low-Reynolds-number lid-driven cavity, DPINN’s centerline velocities agree with Ghia et al. (1982), and its contours and streamlines depict the expected flow pattern.The setup assumes incompressible steady flow at Re = 10 and uses 961 collocation points distributed across ten cells in each direction.Source: §4.3 · pp. 12–14, Figure 8 · p. 13, Figure 7 · p. 13
- The Navier–Stokes demonstration is limited to a low-Reynolds-number lid-driven cavity; higher Reynolds numbers bring unsteady, turbulent, and three-dimensional flow complexity.The authors describe this as a foundation for further direct DPINN solutions of the Navier–Stokes equations.Source: §4.3 · p. 14
5. Conclusions
DPINN partitions the computational domain into cells, using local PINNs and interface constraints to improve representation and data efficiency. The study reports improved representation on an advection test and more accurate, data-efficient Burgers’ solutions, while its Navier–Stokes result is limited to low-Reynolds-number steady states.
5. Conclusions
- DPINN partitions the domain into cells with local PINNs; interface constraints stitch solutions and act as natural regularizers.The authors present this distributed strategy as improving representation capability and data efficiency.Source: §5 · p. 14
- DPINN accurately represents and robustly advects a high-frequency wave packet, whereas PINN fails to represent it.The study presents this as a glimpse of DPINN’s improved representation power.Source: §5 · p. 14
- DPINN yields a more accurate Burgers’ equation solution with less training data than PINN.The paper describes DPINN as more data-efficient than the original PINN.Source: §5 · p. 14
- DPINN solves steady-state Navier–Stokes equations at low Reynolds number; unsteady, full three-dimensional cases remain beyond this study’s scope.The authors say those cases require further analysis, algorithmic advances, and greater computational resources.Source: §5 · p. 14
- The authors identify extending DPINN to inverse problems as a next task.They also mention data-driven discovery of PDEs as future work.Source: §5 · pp. 14–15