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DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks

Shengze Cai, Zhicheng Wang, Lu Lu, Tamer A Zaki, George Em Karniadakis

arXiv:2009.12935v1physics.comp-ph

TL;DR

The paper addresses efficient inference for coupled, multiscale electroconvection fields when only sparse measurements are available. It combines pre-trained DeepONets with a neural-network surrogate in the DeepM&Mnet framework, which accurately infers full 2D fields for unseen electric potentials using few measurements.

  • Problem

    Electroconvection couples flow, electric, and cation/anion concentration fields, creating a multiphysics and multiscale simulation problem with costly small-Debye-length boundary layers and sparse-data assimilation needs.

  • Method

    DeepM&Mnet uses pre-trained DeepONets as fixed operator-based constraints alongside a coordinate-input neural-network surrogate and sparse measurements from any field.

  • Results

    For both parallel and series architectures, only a few measurements were sufficient to infer full electroconvection fields, including cases where some state variables lacked measurements.

  • Takeaways & Limitations

    DeepM&Mnet provides a plug-and-play data assimilation framework using pre-trained DeepONets to build flexible and efficient multiphysics models.

  • Takeaways & Limitations

    The framework uses a fixed electrohydrodynamic coupling constant κ = 0.5 throughout the paper and investigates the L2 regularization term separately.

Abstract

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Electroconvection is a multiphysics problem involving coupling of the flow field with the electric field as well as the cation and anion concentration fields. For small Debye lengths, very steep boundary layers are developed, but standard numerical methods can simulate the different regimes quite accurately. Here, we use electroconvection as a benchmark problem to put forward a new data assimilation framework, the DeepM&Mnet, for simulating multiphysics and multiscale problems at speeds much faster than standard numerical methods using pre-trained neural networks (NNs). We first pre-train DeepONets that can predict independently each field, given general inputs from the rest of the fields of the coupled system. DeepONets can approximate nonlinear operators and are composed of two sub-networks, a branch net for the input fields and a trunk net for the locations of the output field. DeepONets, which are extremely fast, are used as building blocks in the DeepM&Mnet and form constraints for the multiphysics solution along with some sparse available measurements of any of the fields. We demonstrate the new methodology and document the accuracy of each individual DeepONet, and subsequently we present two different DeepM&Mnet architectures that infer accurately and efficiently 2D electroconvection fields for unseen electric potentials. The DeepM&Mnet framework is general and can be applied for building any complex multiphysics and multiscale models based on very few measurements using pre-trained DeepONets in a plug-and-play mode.

1. Introduction

The paper develops DeepONet-based learning for coupled multiphysics fields and introduces DeepM&Mnet to assimilate sparse measurements into full electroconvection solutions. Electroconvection serves as a benchmark for efficient multiphysics and multiscale modeling.

  • Motivation: Deep learning methods have been applied to fluid mechanics, including PDE solution modeling and data-driven identification of unknown equation coefficients.The paper situates its approach alongside physics-informed and other scientific machine-learning methods.
  • DeepONet: DeepONet approximates nonlinear operators by combining a branch net for input functions with a trunk net for output coordinates.This operator-learning structure supports mappings between functions rather than only finite-dimensional inputs and outputs.
  • Electroconvection benchmark: Electroconvection couples mass, momentum, ion transport, and electrostatic fields, while small Debye lengths create costly electric double-layer resolution requirements.The paper notes that simulations commonly use ϵ = 10^-3, whereas physically realistic experiments have ϵ < 10^-4.
  • DeepONet: The authors train separate DeepONets to predict each electroconvection field from inputs drawn from the remaining coupled fields.Unlike image-based CNN mappings, DeepONets do not require all computational-domain grid values in the training data.
  • DeepM&Mnet: DeepM&Mnet integrates pre-trained DeepONets with sparse measurements from any field to infer the full coupled multiphysics system.The framework uses a coordinate-input neural network as the surrogate solution model and DeepONets as solution constraints.
  • DeepM&Mnet: The paper presents parallel and series DeepM&Mnet architectures for general multiphysics and multiscale simulation.The paper is organized around numerical data generation, DeepONet training, DeepM&Mnet architectures, and their evaluation.

2. Numerical simulation of electroconvection and data generation

The study formulates 2D electroconvection using coupled Stokes, electric-potential, and ion-transport equations, then validates a spectral-element solver across applied voltages before generating steady training snapshots.

  • Governing equations: The dimensionless model couples Stokes flow, incompressibility, electric potential, and cation/anion transport equations.The fields include velocity, pressure, electric potential, and ion concentrations; free charge depends on the cation–anion concentration difference.
  • Governing equations: The electrostatic body force is fe = −κρe∇φ/2ϵ^2, with κ = 0.5 and ϵ the Debye length.Here, ρe denotes free charge density and κ is fixed throughout the paper.
  • Boundary conditions: The rectangular domain spans [−3, 3] × [0, 1], with fixed top-wall velocity, ion concentrations, and potential, while the bottom wall is selectively permeable to cations.The bottom boundary is impermeable to anions and maintains c+ = 2; the top boundary imposes c+ = c− = 1 and φ = ∆Φ.
  • Validation simulations: Nine validation simulations used ϵ = 10^-3 and ∆Φ from 25 to 120 on a 48 × 64 clustered spectral-element mesh.The smallest y-direction element was 0.0011, supporting resolution near the walls.
  • Training-data generation: Training data used steady simulations at ϵ = 10^-2 with ∆Φ systematically varying from 5 to 75.Each run stopped at t = 2.5, and snapshots of φ, u, v, and c± were saved; the mesh was reduced to 32 × 32 elements.
  • Validation simulations: Figure 3 compares top-wall current-density dynamics for ∆Φ = 25, 40, 80, and 120 and time-averaged current against prior simulations.The validation uses NekTar results and the simulations of [27].

3. DeepONets for 2D Electroconvection

The paper trains independent DeepONets to map electroconvection fields to one another, using branch and trunk networks to represent input functions and output locations. Training uses NekTar-generated states, tailored losses for multiscale variables, and testing on unseen voltage differences shows accurate, fast predictions.

  • 3.1. The building blocks: DeepONets: Five independent DeepONets learn nonlinear operators that map input fields and coordinates to target electroconvection fields.The networks use a function input U together with location points (x, y) and output G(U)(x, y).
  • 3.1. The building blocks: DeepONets: The architectures use branch networks for discretized input functions and trunk networks for output coordinates.The trunk outputs location-dependent features, while the branch processes sensor values of the input field.
  • 3.1. The building blocks: DeepONets: MSE trains Gφ,c+,c−, whereas MAPE trains Gu,v because velocity fields span a large range of magnitudes.MAPE is preferred when outputs have a large range of function values; η stabilizes the percentage-error expression near zero.
  • 3.2. Training of DeepONets: NekTar supplies 15 steady states with ΔΦ = 5, 10, . . . , 75, represented by 21 × 11 input sensors and 800 output points per state.This produces N = 15 × 800 = 12000 training data points for the supervised DeepONet training.
  • 3.3. Testing of DeepONets: For the unseen case ΔΦ = 62.15, field predictions agree with NekTar references, with MSEs on the order of O(10^-6 −10^-5).The tested networks predict velocity, concentrations, and electric potential using the corresponding input fields.
  • 3.3. Testing of DeepONets: Less than 1 second is required to predict 2D fields on a common GPU after training, yielding about a 10,000-fold speedup over NekTar for independent conditions.Training one DeepONet takes approximately 2 hours, while prediction is a simple network evaluation.

4. DeepM&Mnet for 2D Electroconvection

DeepM&Mnet combines a trainable neural network with fixed, pre-trained DeepONets and sparse measurements to infer coupled 2D electroconvection fields. Parallel and series architectures both produce fields consistent with simulations for unseen conditions.

  • Parallel architecture: DeepM&Mnet uses pre-trained DeepONets as constraints for a trainable neural network solving the coupled electroconvection fields.The loss combines data mismatch, operator mismatch, and L2 regularization; DeepONet outputs constrain the neural-network outputs.
  • Parallel architecture: Measurements are not required for every variable because DeepONets can constrain unmeasured fields from measurements of φ.The framework can use sparse data from any state variable, while operator outputs guide the remaining variables.
  • Series architecture: The series architecture trains only φ while generating u, v, c+, and c− as hidden outputs from fixed DeepONets.With measurements only of φ, the network predicts φ and the embedded operators produce the other fields.
  • 2D testing results: 10 measurements of each field are sufficient for DeepM&Mnet to achieve relative L2-norm errors below 10% for every state variable.Without DeepONet constraints, the comparable neural network requires hundreds of sensors for the same accuracy.
  • 2D testing results: More operator evaluation points improve accuracy, while five randomized training processes provide the reported means and standard deviations.The parallel architecture was evaluated with 20 measurements per condition, and the series architecture used 20 measurements of φ with Nop = 1000.
  • 2D testing results: The series architecture achieves relative L2-norm errors of 0.76% for φ′, 3.75% for u′, 1.72% for v′, 2.70% for c+′, and 1.96% for c−′.These results use only φ measurements and generate the other fields through pre-trained DeepONets.
  • 2D testing results: DeepM&Mnet is presented as a flexible, efficient plug-and-play data-assimilation framework because DeepONets are pre-trained offline and fixed during inference.The framework targets full-field reconstruction from very few measurements.

5. Summary

The paper demonstrates DeepONets for independently predicting coupled electroconvection fields and introduces DeepM&Mnet to reconstruct the full system from sparse measurements. Both parallel and series architectures infer the fields accurately even when measurements do not cover every state variable.

  • Summary: DeepONets independently predict each coupled electroconvection field from inputs consisting of the other fields.The training dataset contains 15 conditions spanning applied electric potentials.
  • Summary: DeepM&Mnet integrates pre-trained DeepONets with a few measurements from any fields to produce the full coupled system.A neural network approximates the multiphysics solution, while fixed DeepONets provide constraints.
  • Summary: Both parallel and series architectures infer full electroconvection fields with few measurements, including cases where some state variables are unmeasured.The DeepONets are trained offline and embedded as reusable building blocks.

Appendix A. DeepONet training: MSE loss vs. MAPE loss

The appendix compares MSE and MAPE losses for the velocity DeepONet, where multiscale velocity magnitudes make relative-error-focused training important. MAPE better captures the low-velocity regime in the reported example.

  • Loss selection: MAPE loss is used for Gu because velocity magnitudes span multiple scales and MSE emphasizes larger-magnitude velocities.Small-scale velocities contribute little to MSE-based optimization.
  • Comparison: At ΔΦ = 13.4, Gu trained with MAPE predicts the flow pattern more accurately than Gu trained with MSE.The velocity magnitude is approximately 10^-4 in this low-potential regime.
  • Comparison: At ΔΦ = 62.15, Gu with MAPE has a relative L2-norm error of 1.65%, despite larger errors than in the low-potential comparison.The velocity magnitude is of order 10^0 in this high-potential regime.

Appendix B. Effect of L2 regularization for DeepM&Mnet

The appendix examines L2 regularization in DeepM&Mnet using 1D electroconvection, where five measurements per coupled field are used to infer profiles. The setup trains reciprocal DeepONets for φ and c−.

  • Regularization: L2 regularization is included to reduce overfitting and stabilize DeepM&Mnet training with sparse data.The DeepM&Mnet loss contains data, operator, and L2 regularization terms.
  • 1D setup: The 1D benchmark models φ and c− as functions of y using reciprocal pre-trained DeepONets.Gφ maps c−(y) to φ(y), while Gc− maps φ(y) to c−(y).
  • 1D setup: The parallel 1D DeepM&Mnet uses five uniformly distributed measurements for each coupled field to infer the profiles at ΔΦ = 77.6.The appendix compares results obtained with different loss functions.
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