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Neural Inverse Operators for Solving PDE Inverse Problems

Roberto MolinaroYunan YangBjörn EngquistSiddhartha Mishra

article2023ICML103 citations

Proposes Neural Inverse Operators, a framework that stacks DeepONet and Fourier Neural Operators to map boundary observation operators directly to PDE coefficients, delivering solutions orders of magnitude faster than traditional PDE-constrained optimization across electrical impedance tomography, wave scattering, optical tomography, and seismic imaging.

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Many critical applications in engineering, geophysics, and medical diagnostics—such as medical tomography, subsurface seismic imaging, and non-destructive material testing—rely on solving partial differential equation (PDE) inverse problems. These problems require inferring internal medium properties (such as tissue conductivity or subsurface rock velocity) solely from indirect boundary or surface measurements. Traditional numerical techniques rely on iterative PDE-constrained optimization or specialized direct algorithms. However, these classical approaches are computationally prohibitive, requiring hours of compute time per sample, and suffer from high sensitivity to initial guesses and measurement noise. While modern data-driven neural operators have accelerated forward PDE simulations, existing deep learning architectures map functions to functions and are fundamentally ill-suited for inverse problems, which mathematically require mapping entire observation operators to interior property fields.

The article designs and evaluates a novel deep learning framework, termed Neural Inverse Operators (NIOs), specifically developed to learn end-to-end inverse mappings from operator measurements directly to underlying spatial fields. The objective is to demonstrate that this architecture can solve challenging, ill-posed PDE inverse problems with higher accuracy, robust generalization, and execution speeds several orders of magnitude faster than both standard deep learning models and traditional numerical inversion techniques.

To achieve this, the authors introduce an architecture that sequentially composes two existing frameworks: Deep Operator Networks (DeepONets) and Fourier Neural Operators (FNOs). The DeepONet component processes boundary-defined measurements and lifts them into internal spatial representations, while the FNO component performs nonlinear mode mixing and final inversion. To ensure the model remains invariant to sample order and input size, the authors introduce a training strategy called randomized batching. The approach was systematically evaluated across four major scientific benchmarks: electrical impedance tomography (the Calderón problem, including a realistic heart-and-lungs model), inverse wave scattering for object detection, optical imaging via radiative transport, and seismic full waveform inversion using standard geophysical datasets. The authors also benchmarked NIO against fully convolutional neural networks, standard DeepONets, traditional direct solvers, and PDE-constrained optimization methods across variations in noise, sensor placement, grid resolution, and out-of-distribution distributions.

The experimental findings show that NIO consistently matches or outperforms all baseline models across every benchmark, often cutting error rates substantially. In electrical impedance tomography, NIO reduced relative L1 reconstruction errors to 0.85% (compared to 1.49% for convolutional baselines and 8.75% for classical direct methods). In inverse wave scattering, NIO achieved a 1.07% median test error compared to 2.53% for convolutional networks. When tested against traditional PDE-constrained optimization, NIO proved to be roughly five times more accurate (2.3% error versus 11.1%) while running in less than one second on a standard CPU—achieving an inference speedup of approximately four orders of magnitude compared to the 8.5 hours required by GPU-based optimization. Furthermore, unlike convolutional baselines whose errors degraded sharply when the number of input measurements fluctuated, NIO remained highly stable across varying measurement counts, sensor locations, grid resolutions, and input noise levels.

These results demonstrate that NIO provides an operationally viable, robust path to real-time imaging and physical property estimation. By replacing expensive iterative solvers with an ultra-fast, direct inference pipeline, organizations can dramatically lower compute costs, accelerate turnaround times from hours to fractions of a second, and make real-time operational decisions in time-critical environments like bedside patient monitoring and hazard assessment.

Based on these outcomes, organizations working on complex inverse physical systems should consider piloting neural inverse operators as high-speed replacements for iterative PDE inversion pipelines. Prior to deployment, teams should validate model robustness against domain-specific measurement noise and establish training data generation protocols using representative forward simulations. For future research, the authors recommend extending NIO to higher-dimensional 3D seismic settings, adapting alternative operator backbones (such as graph neural networks), and developing rigorous theoretical approximation bounds to formalize the architecture's mathematical guarantees.

While confidence in the reported experimental performance is high due to comprehensive benchmarking, practical limitations remain. The current empirical validation is primarily restricted to one- and two-dimensional synthetic and benchmark datasets with modest measurement counts (typically 20 to 32 boundary measurements during training). Real-world field deployments may introduce higher levels of unstructured noise, missing channels, and severe three-dimensional scaling challenges that warrant caution and targeted pilot validation before full-scale adoption.

inverse problemsdiffusion equationsNeural inverse operatorsPDE inverse problemsoperator learning
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Topics

inverse problemsdiffusion equations

Keywords

Neural inverse operatorsPDE inverse problemsoperator learningAlbedo operatorboundary observation operatorFourier layers

Table of Contents

  • 1. Introduction
  • 2. A Class of Inverse Problems
  • 2.1. Mathematical Framework
  • 2.2. Calderón Problem (EIT)
  • 2.3. Inverse Wave Scattering
  • 2.4. Radiative Transport and Optical Imaging
  • 2.5. Seismic Imaging
  • 3. Neural Inverse Operators
  • 3.1. Learning Task and Challenges
  • 3.2. Existing Operator Learning Architectures
  • 3.3. A Motivating (Formal) Calculation
  • 3.4. The Architecture
  • 4. Empirical Results
  • 4.1. Calderón Problem for EIT
  • 4.2. Inverse Wave Scattering
  • 4.3. Radiative Transport Equation and Optical Imaging
  • 4.4. Seismic Imaging
  • 4.5. Robustness and Computational Efficiency of NIO
  • 5. Related Work
  • 6. Discussion
  • Acknowledgements
  • References
  • A. Depiction of PDE Inverse Problems
  • B. Proof of Formula (23) in Main Text
  • C. Mathematical Description of Heart and Lungs Phantom
  • D. Architecture and Training Details
  • D.1. Architecture Details
  • D.1.1. Feed Forward Dense Neural Networks
  • D.1.2. Fully Convolutional Neural Network
  • D.1.3. DeepONet
  • D.1.4. Fourier Neural Operator
  • D.1.5. Neural Inverse Operator
  • D.2. Training Details
  • D.2.1. Sensitivity to Initialization of the Trainable Parameters
  • D.2.2. Sensitivity to the Number of Training Samples
  • E. Further Experimental Results
  • E.1. Illustration of Results Reported in Table 1
  • E.2. Robustness of Reconstruction to Λₐ-Discretization
  • E.3. Robustness of Reconstruction to Noise
  • E.4. Robustness of Reconstructions to Varying Grid Sizes
  • E.5. Robustness of Reconstruction to Random Sensor Location
  • E.6. Out-of-Distribution Reconstruction
  • E.7. Ablation Studies
  • E.8. Comparison with Standard Numerical Methods for Inverse Problems
  • E.8.1. Calderón Problem: Heart and Lungs
  • E.8.2. Inverse Wave Scattering
  • E.8.3. Seismic Imaging

Knowls

  1. Knowl 1 — PDE inverse problems as operator-to-function maps

    definition

    Let a be an unknown spatial coefficient in a PDE posed on a domain D. For each admissible boundary input g, solving the PDE and measuring the solution defines a boundary observation operator Λa:g↦h(u)\Lambda_a:g\mapsto h(u), where uu is the PDE solution and h(u)h(u) is the observable. The inverse task is to recover the coefficient from that entire operator: F−1:Λa↦aF^{-1}:\Lambda_a\mapsto a. Thus, the input is an operator represented by boundary input-output measurements, while the output is a function defined in the domain interior. The paper considers inverse problems for which this operator-to-function formulation is the well-defined one, rather than attempting to infer a from only one or a few individual PDE solutions.

  2. Knowl 2 — Neural Inverse Operator architecture

    model/method

    A Neural Inverse Operator (NIO) maps a collection of boundary measurements to an interior coefficient by composing a DeepONet with a Fourier neural operator (FNO). For each measurement Ψℓ=Λa(gℓ)\Psi_\ell=\Lambda_a(g_\ell) and interior location zz, the DeepONet branch processes the boundary measurement and its trunk processes zz, producing an interior representation fℓ(z)f_\ell(z). The representations are aggregated by the lifting map

    h(z)=D(1L∑ℓ=1Lfℓ(z))+Ez,h(z)=D\left(\frac{1}{L}\sum_{\ell=1}^{L}f_\ell(z)\right)+Ez,

    where LL is the number of measurement samples, DD and EE are learned linear maps into the FNO lifting dimension, and zz is the spatial coordinate. The nonlinear FNO component then processes the field hh and outputs the reconstructed coefficient a∗(z)a^*(z). The NIO input consists of the locations and the measurements {Ψℓ}\{\Psi_\ell\}; it does not require the paired boundary conditions gℓg_\ell. The averaging makes the aggregation insensitive to the order of the measurement samples, while the FNO performs nonlinear mixing of the resulting channels.

  3. Knowl 3 — Randomized batching for variable numbers of measurements

    algorithm

    NIO is trained with randomized batching so that it can process different numbers of boundary-measurement samples at training and test time. Given LL available measurements for one training example, each training iteration draws a batch size LbL_b from {2,…,L}\{2,\ldots,L\}, randomly selects LbL_b of the measurements, and feeds that subset to the NIO. The model aggregates the selected representations by their average and predicts the coefficient. Repeating this procedure exposes the model to subsets of varying sizes and orderings rather than only the fixed-size measurement set. At inference, the same architecture can be evaluated directly on a different number of measurements without changing its structure.

  4. Knowl 4 — Benchmark problems and experimental data

    experimental setup

    The experiments evaluate NIO on six PDE inverse-problem benchmarks. For electrical impedance tomography, the Calderón problem uses a conductivity on [0,1]2[0,1]^2 sampled as the exponential of a sum of sine modes, with between one and five modes and random coefficients; each example supplies 20 Dirichlet-to-Neumann measurements. A second Calderón benchmark uses a heart-and-lungs conductivity phantom on the unit circle, with 8% random perturbations to phantom parameters and Fourier boundary inputs. In inverse wave scattering, the Helmholtz equation on [0,1]2[0,1]^2 is used to recover media containing one to four randomly located square-shaped inclusions, from 20 Dirichlet-to-Neumann measurements. In optical imaging, the radiative transport equation on [0,1]×[−1,1][0,1]\times[-1,1] is used to infer a discontinuous absorption profile; 32 inflow measurements form the Albedo-operator input. Seismic imaging uses the acoustic wave equation and the CurveVel-A and Style-A coefficient datasets, with five source locations and time-dependent receiver measurements.

    For the trigonometric Calderón, inverse-scattering, and optical-imaging benchmarks, the models use 4096 training examples; their test sets contain 2048, 1600, and 2048 examples, respectively. Seismic models use 22,000 training and 6,000 test examples for CurveVel-A, and 55,000 training and 7,000 test examples for Style-A. PDE data are generated numerically using finite-difference or finite-element solvers as appropriate. The comparison models are a DeepONet with a convolutional branch (DONet) and a fully convolutional neural network (FCNN).

  5. Knowl 5 — NIO accuracy across the six benchmarks

    data/table

    The table reports relative median test errors for DONet, FCNN, and NIO; all values are percentages. NIO has the lowest error in both norms on the four non-seismic benchmarks and on both seismic benchmarks, where it is slightly better than FCNN. The results show that the operator-to-function architecture is effective across elliptic, wave, transport, and seismic inverse problems.

    Benchmark DONet L1L^1 DONet L2L^2 FCNN L1L^1 FCNN L2L^2 NIO L1L^1 NIO L2L^2
    Calderón, trigonometric 1.97% 2.36% 1.49% 1.82% 0.85% 1.05%
    Calderón, heart and lungs 0.95% 3.69% 0.27% 1.62% 0.18% 1.16%
    Inverse wave scattering 3.83% 7.41% 2.53% 7.55% 1.07% 2.94%
    Radiative transport 2.35% 4.35% 1.46% 3.71% 1.1% 2.9%
    Seismic imaging, CurveVel-A 3.98% 5.86% 2.65% 5.05% 2.71% 4.71%
    Seismic imaging, Style-A 3.82% 5.17% 3.12% 4.63% 3.04% 4.36%

  6. Knowl 6 — Robustness to measurement count and discretization

    empirical result

    NIO was tested with a different number of boundary measurements from the number used during training. In the paper's experiments, training typically used 20–32 measurements; tests included fewer measurements and, for the trigonometric Calderón and inverse-scattering problems, newly generated inputs with 100 measurements. NIO's error changed only slightly as the measurement count decreased, whereas the DONet and FCNN baselines deteriorated substantially, in some comparisons by nearly an order of magnitude. The result supports the claim that NIO learns from the operator represented by the measurements, rather than relying on a fixed-size discrete input.

    NIO also retained accuracy when the input grid resolution or sensor locations changed. For inverse scattering, a model trained at 70×7070\times70 resolution had relative median L1L^1 errors of 0.93% at 50×5050\times50 and 0.95% at 100×100100\times100. With 200 randomly placed boundary sensors, rather than the 272 equispaced training sensors, the relative median L1L^1 errors were 1.18% for trigonometric Calderón and 1.43% for inverse scattering.

    With 1% noise added to test inputs, NIO's relative median L1L^1 errors were 0.91% (trigonometric Calderón), 0.18% (heart-and-lungs Calderón), 3.72% (inverse scattering), 1.1% (radiative transport), 2.73% (CurveVel-A), and 3.09% (Style-A). These measurements quantify robustness but also show that noise impact varies by benchmark.

  7. Knowl 7 — Generalization to out-of-distribution coefficients

    empirical result

    The paper tested whether models trained on the benchmark distributions could reconstruct coefficients from changed distributions. The trigonometric Calderón tests used up to six frequency modes, beyond the training range, with two different decays of the higher modes. The heart-and-lungs test increased phantom-parameter perturbations from 8% during training to 12%. The inverse-scattering tests used either a fixed five inclusions or one to four inclusions with varying shapes. The table gives relative median L1L^1 errors in percent. NIO had the lowest error in all five tests; its errors rose on some shifted distributions but remained within about four times the corresponding in-distribution errors, as reported by the authors.

    Out-of-distribution test DONet FCNN NIO
    Trigonometric Calderón, higher-frequency distribution A 1.37% 1.27% 1.2%
    Trigonometric Calderón, higher-frequency distribution B 1.62% 1.28% 0.91%
    Heart-and-lungs Calderón, 12% perturbations 1.05% 0.28% 0.19%
    Inverse scattering, five inclusions 4.61% 3.84% 3.0%
    Inverse scattering, varying inclusion shapes 8.61% 8.98% 4.54%

  8. Knowl 8 — Ablations identify the roles of nonlinear mixing and randomized batching

    empirical result

    Two NIO components were ablated while keeping the selected benchmark hyperparameters fixed. First, the nonlinear FNO component was removed and the lifting dimension set to one, leaving only the linear average of the DeepONet representations. This increased generalization error by nearly a factor of two for the trigonometric Calderón problem and by as much as a factor of six for inverse wave scattering. Second, randomized batching was removed. Its inclusion improved results by factors of roughly 5–10 for the Calderón and radiative-transport problems, while its benefit was more modest for inverse scattering. The authors note that the training sets contained only 20–32 measurements per operator, so the limited measurement count may constrain the gains from nonlinear mixing.

  9. Knowl 9 — Accuracy and speed relative to conventional inverse methods

    empirical result

    NIO was compared with direct inversion and PDE-constrained optimization on selected cases. For the heart-and-lungs Calderón problem, the D-bar direct method had an L1L^1 test error of 8.75%, compared with approximately 0.15% for NIO; the D-bar method took about two hours per sample, while NIO inference took 0.1 seconds on a CPU. For an out-of-distribution inverse-scattering example, NIO's L1L^1 error was 2.3%, versus 11.1% for PDE-constrained optimization, with inference under one second on a CPU compared with 8.5 hours for the optimization run. For two CurveVel-A seismic examples, NIO errors were 2.03% and 4.84%, compared with 5.05% and 16.9% for PDE-constrained optimization; the latter took about 30 minutes on an 8-core M1 CPU. These experiments show large speed and accuracy advantages in the reported comparisons, not a universal runtime guarantee.

  10. Knowl 10 — Limitations and open theoretical questions

    limitation

    The paper provides empirical evidence for NIO but does not establish approximation bounds, universality results, or other theoretical guarantees for the architecture. It also identifies scaling to larger problem sizes, particularly higher-dimensional seismic imaging, as an open experimental question. The reported performance and robustness therefore apply to the tested benchmarks and resolutions; the paper does not establish how NIO scales to substantially higher dimensions or arbitrary inverse-problem distributions.

Coverage note — The formal Helmholtz eigenfunction calculation is omitted as a separate knowl because it serves as heuristic motivation for the architecture rather than an independent result; detailed hyperparameter-search settings are also omitted as implementation detail.

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Citation

MLA
Molinaro, R., et al. “Neural Inverse Operators for Solving PDE Inverse Problems”. International Conference on Machine Learning, vol. 202, 2023, pp. 25105–39, https://proceedings.mlr.press/v202/molinaro23a.html.
APA
Molinaro, R., Yang, Y., Engquist, B., & Mishra, S. (2023). Neural Inverse Operators for Solving PDE Inverse Problems. International Conference on Machine Learning, 202, 25105–25139. https://proceedings.mlr.press/v202/molinaro23a.html
Chicago
Molinaro, R., Y. Yang, B. Engquist, and S. Mishra. 2023. “Neural Inverse Operators for Solving PDE Inverse Problems”. International Conference on Machine Learning 202: 25105–39. https://proceedings.mlr.press/v202/molinaro23a.html.
Harvard
Molinaro, R. et al. (2023) “Neural Inverse Operators for Solving PDE Inverse Problems”, International Conference on Machine Learning. PMLR, pp. 25105–25139. Available at: https://proceedings.mlr.press/v202/molinaro23a.html.
Vancouver
1. Molinaro R, Yang Y, Engquist B, Mishra S (2023) Neural Inverse Operators for Solving PDE Inverse Problems. In: International Conference on Machine Learning. PMLR, pp 25105–25139

BibTeX

@InProceedings{pmlr-v202-molinaro23a,
  title = 	 {Neural Inverse Operators for Solving {PDE} Inverse Problems},
  author =       {Molinaro, Roberto and Yang, Yunan and Engquist, Bj\"{o}rn and Mishra, Siddhartha},
  booktitle = 	 {Proceedings of the 40th International Conference on Machine Learning},
  pages = 	 {25105--25139},
  year = 	 {2023},
  editor = 	 {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan},
  volume = 	 {202},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {23--29 Jul},
  publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v202/molinaro23a/molinaro23a.pdf},
  url = 	 {https://proceedings.mlr.press/v202/molinaro23a.html},
  abstract = 	 {A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.}
}
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