The authors combine H(div)-L2 subspaces from Raviart-Thomas and dgP0 elements with a transformer and GP regression on fluxes to create real-time structure-preserving surrogates with closed-form posterior uncertainty for Dirichlet-to-Neumann maps.
Solving forward and inverse pde problems on unknown manifolds via physics-informed neural operators
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abstract
In this paper, we evaluate the effectiveness of deep operator networks (DeepONets) in solving both forward and inverse problems of partial differential equations (PDEs) on unknown manifolds. By unknown manifolds, we identify the manifold by a set of randomly sampled data point clouds that are assumed to lie on or close to the manifold. When the loss function incorporates the physics, resulting in the so-called physics-informed DeepONets (PI-DeepONets), we approximate the differentiation terms in the PDE by an appropriate operator approximation scheme. For the second-order elliptic PDE with a nontrivial diffusion coefficient, we approximate the differentiation term with one of these methods: the Diffusion Maps (DM), the Radial Basis Functions (RBF), and the Generalized Moving Least Squares (GMLS) methods. For the GMLS approximation, which is more flexible for problems with boundary conditions, we derive the theoretical error bound induced by the approximate differentiation. Numerically, we found that DeepONet is accurate for various types of diffusion coefficients, including linear, exponential, piecewise linear, and quadratic functions, for linear and semi-linear PDEs with/without boundaries. When the number of observations is small, PI-DeepONet trained with sufficiently large samples of PDE constraints produces more accurate approximations than DeepONet. For the inverse problem, we incorporate PI-DeepONet in a Bayesian Markov Chain Monte Carlo (MCMC) framework to estimate the diffusion coefficient from noisy solutions of the PDEs measured at a finite number of point cloud data. Numerically, we found that PI-DeepONet provides accurate approximations comparable to those obtained by a more expensive method that directly solves the PDE on the proposed diffusion coefficient in each MCMC iteration.
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2026 6representative citing papers
HilbNets define convolutions via Hilbert bundle connection Laplacians, prove that sampled Hilbert cellular sheaf Laplacians converge to the continuous operator, and show that discretized networks are consistent and transferable across samplings.
A nested Fourier-MIONet surrogate predicts radiative heat transfer in multi-resolution 3D fire simulations with 2-4% error at reduced computational cost compared to direct RTE solves.
LESnets integrates LES equations and the law of the wall into F-FNO to enable data-free, stable long-term predictions of wall-bounded turbulence at Re_tau up to 1000 on coarse grids, matching traditional LES accuracy at higher efficiency.
NOTES couples a DeepONet topology decoder with CMA-ES in a PCA-derived latent space, achieving >95% deflection efficiency on nanophotonic metagratings and compliance of 246 on MBB beams, outperforming direct CMA-ES and gradient-based baselines.
PINNs and DeepONets solve Newtonian plane Couette flow with dynamic wall slip; DeepONet achieves 0.36% mean relative error on unseen cases and 540X speedup over numerical methods.
citing papers explorer
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Structure-Preserving Neural Surrogates with Tractable Uncertainty Quantification
The authors combine H(div)-L2 subspaces from Raviart-Thomas and dgP0 elements with a transformer and GP regression on fluxes to create real-time structure-preserving surrogates with closed-form posterior uncertainty for Dirichlet-to-Neumann maps.
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Consistent Geometric Deep Learning via Hilbert Bundles and Cellular Sheaves
HilbNets define convolutions via Hilbert bundle connection Laplacians, prove that sampled Hilbert cellular sheaf Laplacians converge to the continuous operator, and show that discretized networks are consistent and transferable across samplings.
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Nested Fourier-enhanced neural operator for efficient modeling of radiation transfer in fires
A nested Fourier-MIONet surrogate predicts radiative heat transfer in multi-resolution 3D fire simulations with 2-4% error at reduced computational cost compared to direct RTE solves.
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Large-eddy simulation nets (LESnets) based on physics-informed neural operator for wall-bounded turbulence
LESnets integrates LES equations and the law of the wall into F-FNO to enable data-free, stable long-term predictions of wall-bounded turbulence at Re_tau up to 1000 on coarse grids, matching traditional LES accuracy at higher efficiency.
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Neural Operator-enabled Topology-informed Evolutionary Strategy for PDE-Constrained Optimization
NOTES couples a DeepONet topology decoder with CMA-ES in a PCA-derived latent space, achieving >95% deflection efficiency on nanophotonic metagratings and compliance of 246 on MBB beams, outperforming direct CMA-ES and gradient-based baselines.
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Solution of the Newtonian plane Couette flow with dynamic wall slip using machine-learning methods
PINNs and DeepONets solve Newtonian plane Couette flow with dynamic wall slip; DeepONet achieves 0.36% mean relative error on unseen cases and 540X speedup over numerical methods.