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Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport
Lu Lu, Raphael Pestourie, Steven G. Johnson, Giuseppe Romano
TL;DR
DeepONet training often requires large high-fidelity datasets that are difficult to obtain. The paper develops a multifidelity DeepONet that combines residual learning and input augmentation, achieving substantially lower error with the same high-fidelity data and supporting fast inverse design for nanoscale heat transport. Its design-search outputs require verification because surrogate error can make the nominal best design differ from the high-fidelity optimum.
Problem
Training DeepONet usually requires large high-fidelity datasets, which are often difficult to obtain in engineering problems.
Method
The multifidelity DeepONet couples two DeepONets through residual learning and input augmentation to use datasets with different fidelity levels.
Results
Multifidelity DeepONet achieves one order of magnitude smaller error than single-fidelity DeepONet using the same amount of high-fidelity data.
Takeaways & Limitations
Combining the trained multifidelity DeepONet with genetic algorithms or topology optimization enables fast inverse design for multiple objective functions without retraining.
Takeaways & Limitations
Because the multifidelity DeepONet has a 3.34% error, several candidate designs must be verified with the high-fidelity numerical solver.
Abstract
from arXiv · showhide
Deep neural operators can learn operators mapping between infinite-dimensional function spaces via deep neural networks and have become an emerging paradigm of scientific machine learning. However, training neural operators usually requires a large amount of high-fidelity data, which is often difficult to obtain in real engineering problems. Here, we address this challenge by using multifidelity learning, i.e., learning from multifidelity datasets. We develop a multifidelity neural operator based on a deep operator network (DeepONet). A multifidelity DeepONet includes two standard DeepONets coupled by residual learning and input augmentation. Multifidelity DeepONet significantly reduces the required amount of high-fidelity data and achieves one order of magnitude smaller error when using the same amount of high-fidelity data. We apply a multifidelity DeepONet to learn the phonon Boltzmann transport equation (BTE), a framework to compute nanoscale heat transport. By combining a trained multifidelity DeepONet with genetic algorithm or topology optimization, we demonstrate a fast solver for the inverse design of BTE problems.
1 Introduction
Deep neural operators have emerged as a SciML paradigm for learning mappings between infinite-dimensional function spaces, but DeepONet training often needs extensive high-fidelity data. This work addresses that data challenge with multifidelity learning and applies the resulting model to fast inverse design of nanoscale heat transport.
- Deep neural operators learn mappings between infinite-dimensional function spaces for PDE surrogate solving.
- DeepONet has demonstrated applications across diverse scientific and engineering problems and may overcome the curse of dimensionality for some PDEs.
- Training DeepONet usually requires substantial high-fidelity data, which can be difficult to obtain in engineering problems.
- The proposed multifidelity DeepONet fuses datasets with different fidelity levels using residual learning and input augmentation.
- Multifidelity DeepONet is applied to phonon BTE heat transport and combined with genetic algorithms or topology optimization for inverse material design.
2 Methods
The methods section introduces DeepONet, multifidelity learning approaches, and the use of trained models for inverse design.
- The paper presents DeepONet, multifidelity learning approaches, and inverse-design procedures using trained DeepONet models.
2.1 DeepONet
DeepONet learns a nonlinear operator from input and output functions using branch and trunk subnetworks. The branch processes sampled input-function values, while the trunk processes the output location.
- DeepONet learns nonlinear operators mapping an input function to an output function, such as boundary or initial conditions to a PDE solution.
- The branch net takes sampled evaluations of the input function, while the trunk net takes the location where the output is predicted.
- The DeepONet output is formed from an inner product of the branch and trunk outputs with a bias term.
2.2 Multifidelity DeepONet
Multifidelity DeepONet combines a large, inexpensive low-fidelity dataset with a smaller high-fidelity dataset to learn accurate operators. It uses residual learning and input augmentation, with their combination improving accuracy in the reported experiments.
- Multifidelity learning: Multifidelity learning combines a small high-fidelity dataset with a larger, cheaper low-fidelity dataset.The high-fidelity dataset has size NH, the low-fidelity dataset has size NL, and NH ≪ NL.
- Residual learning: Residual learning trains one DeepONet on the low-fidelity operator and another on the difference between high- and low-fidelity outputs.The low-fidelity solution captures the basic trend, while the second DeepONet learns the residual operator.
- Input augmentation: Input augmentation supplies the low-fidelity prediction as an additional input to the high-fidelity DeepONet.Approach I appends the entire low-fidelity function to branch inputs, whereas Approach II appends its value at the prediction location to trunk inputs.
- Input augmentation: Approach II achieves better accuracy than Approach I in the Poisson-equation experiments.
- Combined architecture: The combined multifidelity architecture uses a low-fidelity DeepONet, a residual-learning high-fidelity DeepONet, and low-fidelity predictions as trunk inputs.
- Error decomposition: The multifidelity prediction error depends on both the low-fidelity DeepONet error and the residual high-fidelity DeepONet error.
2.3 Inverse design
The learned operator serves as a fast surrogate for inverse design, enabling optimization over material or design variables with either genetic algorithms or gradient-based topology optimization.
- A surrogate operator G enables predictions for any input v within a fraction of a second, supporting inverse design as an optimization problem.
- Genetic algorithm: Genetic algorithms evolve randomly generated candidate solutions through selection, mutation, and crossover across generations.
- Gradient-based topology optimization: Topology optimization treats each pixel as a degree of freedom and iteratively updates material properties with gradient-based optimization.
- Gradient-based topology optimization: Topology optimization uses continuous relaxation, allowing differentiable evaluations and gradients between binary material inputs before producing a binary final design.
- Gradient-based topology optimization: A smoothed Heaviside thresholding function progressively pushes design values toward binary states as β increases, with a penalty added if binarization fails.
3 Results
The study applies multifidelity DeepONet to Poisson and Boltzmann transport problems, then uses the learned operator for inverse design of thermal transport in nanostructures.
- The proposed multifidelity DeepONet is applied to the Poisson equation and Boltzmann transport equation, followed by inverse design of thermal transport in nanostructures.
- The study’s multifidelity DeepONet codes are implemented with DeepXDE and are intended for deposit in GitHub.
3.1 Poisson equation
The Poisson experiment compares multifidelity DeepONet variants using a deliberately small, sparsely observed high-fidelity dataset. Residual learning provides the strongest reported improvement, while input augmentation effects depend on the approach and combination.
- Problem setup: The experiment learns the operator G: f 7→u for a 1D Poisson equation with homogeneous Dirichlet boundary conditions.
- Problem setup: High-fidelity solutions use Δx = 1/99, whereas low-fidelity solutions use Δx = 1/9; the low-fidelity solver has m.s.e. 0.0233.
- Problem setup: The high-fidelity dataset contains 500 forcing samples but only one randomly sampled solution value per forcing function.
- High-fidelity baseline: 0.0157 ± 0.0069 is the smallest test m.s.e. for the high-fidelity-only DeepONet, obtained with width 5 over 10 runs.
- Multifidelity variants: Residual learning reduces error by about one order of magnitude relative to high-fidelity-only DeepONet, reaching 0.0021 ± 0.0007 at width 4.
- Multifidelity variants: Input augmentation approach II reaches 0.0052 ± 0.0033 at width 5 and is uniformly better than the high-fidelity-only model, while approach I shows no improvement.
- Multifidelity variants: Combining residual learning with input augmentation approach II yields 0.0031 ± 0.0006 at width 5, but the reported improvement depends on the combination setting.
3.2 Boltzmann transport equation
The study applies multifidelity DeepONet to steady-state phonon BTE heat transport in nanoporous materials, using pore geometries as inputs and flux fields as outputs. Low-fidelity data and multifidelity strategies reduce errors relative to high-fidelity-only learning, while predictions are evaluated against BTE simulations.
- Problem setup: The BTE models nondiffusive heat transport when a semiconductor’s characteristic length approaches the phonon mean-free-path.The problem uses a rectangular domain with periodic boundaries, a 1 K temperature difference, and pore locations arranged on a five-by-five grid.
- Problem setup: Each dataset sample pairs pore locations with the corresponding flux solution at mesh nodes, using five source iterations for high-fidelity data and two for low-fidelity data.There are approximately 3.4×10^7 possible pore configurations, with about 902 mesh nodes per geometry on average.
- Baseline comparison: The low-fidelity solver has m.s.e. 5.09×10−3 relative to the high-fidelity solution, while high-fidelity-only DeepONet requires about 150 training data to reach comparable accuracy.These are the two baselines used for evaluating multifidelity DeepONet on BTE.
- Effectiveness of multifidelity learning: Residual learning and input augmentation approach II each improve BTE accuracy by one order of magnitude, with their relative advantage depending on dataset size.Residual learning is better below 100 samples, input augmentation becomes better otherwise, and combining both approaches yields still better accuracy.
- Multifidelity DeepONet: A multifidelity DeepONet trained with 1,000 high-fidelity samples achieves testing m.s.e. 8.89±0.07×10−5 and L2 relative error 3.34±0.01%.Replacing the low-fidelity DeepONet with the exact low-fidelity solver gives testing m.s.e. 6.00±0.04×10−5 and L2 relative error 2.72±0.01%.
- Multifidelity DeepONet: Two examples compare reference BTE solutions, multifidelity DeepONet predictions, and pointwise absolute errors for identical pore geometries within each row.The figure presents the reference, prediction, and error fields side by side.
3.3 Boltzmann transport equation for inverse design
The trained multifidelity DeepONet serves as a surrogate for inverse thermal-transport design, supporting genetic algorithms and topology optimization under several heat-flux objectives and pore constraints. Its predictions are checked against high-fidelity numerical solutions, while surrogate error motivates verifying multiple candidate designs.
- Inverse-design setup: The multifidelity DeepONet is used as a surrogate model for optimizing thermal-transport objective functions with genetic algorithms and topology optimization.The same surrogate supports different objective functions and constraints without retraining.
- Continuous optimization: Topology optimization treats binary pore variables as continuous values on [0, 1], but predictions for intermediate values are not guaranteed because training used only binary inputs.An example shows reasonable intermediate predictions, while the authors retain this general qualification.
- Single-point objective: ∼2.4 final objective values were achieved similarly by GA and TO for center heat-flux maximization.For TO, abrupt curve jumps reflect final evaluation after fully binarizing the converged structure.
- Single-point objective: 3.34% surrogate error motivates evaluating the top 6 candidates with the high-fidelity numerical solver rather than trusting only the surrogate-ranked best design.For center optimization, the solver identifies Fig. 8E as best with Jopt ≈2.343; the surrogate ranked it third, while the top two were 1.58% and 2.18% worse.
- Multiple-point objective: Jopt ≈ 1.987 was obtained for two-point heat-flux maximization, with the multifidelity-selected Fig. 9D also best under numerical verification.GA and TO reached similar objective values of ∼2, and most candidate relative differences were within 3%.
- Pore-count constraints: Stronger pore-count constraints produce smaller objective values, while best-design pore locations form nested subsets across N = 11, 9, and 7.The constrained cases are evaluated for N = 11, N = 9, and N = 7.
4 Conclusion
The study develops a multifidelity DeepONet that combines two DeepONets through residual learning and input augmentation. It reduces error relative to single-fidelity training and supports reuse for fast inverse design.
- A multifidelity DeepONet couples two independent DeepONets through residual learning and input augmentation.
- One order of magnitude smaller error is achieved than with single-fidelity DeepONet using the same amount of high-fidelity data.
- The trained network can be reused with genetic algorithms or topology optimization for fast inverse design of multiple objective functions.
- Approach II, which uses a low-fidelity prediction at one point, is more accurate than Approach I, which uses the entire low-fidelity function.
- The sub-DeepONets are vanilla architectures, and newer DeepONet extensions may further improve multifidelity performance.
A Best designs in Section 3.3.4
Table 4 evaluates designs that maximize normalized heat flux at multiple points while constraining the number of pores. It compares objective values computed by the multifidelity DeepONet and numerical solver.
- The designs maximize normalized heat flux on multiple points.
- The optimization imposes a constraint on the number of pores.
- Objective values are computed using both the multifidelity DeepONet and the numerical solver.