arXiv subjects

Lianghao Cao

Publications and source records attributed to Lianghao Cao.

12 recordsLinked to original sources

arXiv2026

On Prior-to-Posterior Stability in the Wasserstein Metric for Bayesian Inverse Problems

Priors in Bayesian inverse problems are often approximated through discretization, hyperparameter estimation, or generative modeling. Understanding how prior approximation errors propagate to the posterior and subsequent predictions is therefore important. In this work, we study the stability of the prior-to-posterior map where both prior and posterior perturbations are measured in the same Wasserstein metric $W_p$, $p\geq1$. We identify verifiable conditions on likelihood regularity and admissible prior classes that ensure uniform, Hölder, and Lipschitz stability. For bounded likelihoods that are uniformly continuous on bounded sets, uniform stability holds over prior classes with uniformly integrable $p$-th moments and a common positive evidence lower bound. With global Hölder regularity of the likelihood and uniform bounds on higher prior moments, a coupling argument leads to a Hölder estimate with a sharp exponent. For $p>1$, a Lipschitz likelihood need not give Lipschitz stability, even for priors with bounded support. We establish Lipschitz stability through an interpolation argument under uniform Poincaré bounds and a globally Lipschitz potential with uniformly bounded essential oscillation under the priors. For Gaussian priors with additive Gaussian noise and bounded Lipschitz forward models, these estimates give posterior $W_2$ bounds even for mutually singular prior perturbations. Numerical experiments for a Darcy inverse problem illustrate the predicted Hölder and Lipschitz rates and the resulting control of errors in the posterior mean and standard deviation of a Lipschitz quantity of interest.

math.NA↗

arXiv2026

A Neural-Network Framework to Learn History-Dependent Constitutive Laws and Identifiability of Internal Variables

The identification of constitutive laws is ubiquitous in engineering: in modeling of materials where experimental data are fitted to mathematical models or learning surrogate models to beat the FE\textsuperscript{2} computational cost of multiscale numerical simulations. However, these models of constitutive laws, unless equipped with a potential formulation, are not necessarily consistent with (a) the second law of thermodynamics; (b) stability of the material under extreme applied strain; and (c) the mathematical theory underpinning the existence of solutions of the governing equation. In this work, we present a causal and energetic formulation, consistent with aforementioned properties, of learning a history-dependent constitutive law. This characterization of the class of internal variables sheds light on the equivalence class of equivalent surrogate models for the constitutive law. We show that the internal variables that are learned from the data are unique up to a linear transform. The framework is deployed to learn the Taylor-averaged response of a polycrystalline magnesium unit cell. We achieve 2\% relative error in the prediction of the Taylor-averaged response.

cond-mat.mtrl-sci↗

arXiv2026

Optimal Experimental Design for Reliable Learning of History-Dependent Constitutive Laws

History-dependent constitutive models serve as macroscopic closures for the aggregated effects of micromechanics. Their parameters are typically learned from experimental data. With a limited experimental budget, eliciting the full range of responses needed to characterize the constitutive relation can be difficult. As a result, the data can be well explained by a range of parameter choices, leading to parameter estimates that are uncertain or unreliable. To address this issue, we propose a Bayesian optimal experimental design framework to quantify, interpret, and maximize the utility of experimental designs for reliable learning of history-dependent constitutive models. In this framework, the design utility is defined as the expected reduction in parametric uncertainty or the expected information gain. This enables in silico design optimization using simulated data and reduces the cost of physical experiments for reliable parameter identification. We introduce two approximations that make this framework practical for advanced material testing with expensive forward models and high-dimensional data: (i) a Gaussian approximation of the expected information gain, and (ii) a surrogate approximation of the Fisher information matrix. The former enables efficient design optimization and interpretation, while the latter extends this approach to batched design optimization by amortizing the cost of repeated utility evaluations. Our numerical studies of uniaxial tests for viscoelastic solids show that optimized specimen geometries and loading paths yield image and force data that significantly improve parameter identifiability relative to random designs, especially for parameters associated with memory effects.

cond-mat.mtrl-sci↗

arXiv2026

Derivative-Informed Fourier Neural Operator: Universal Approximation and Applications to PDE-Constrained Optimization

We present approximation theories and efficient training methods for derivative-informed Fourier neural operators (DIFNOs) with applications to PDE-constrained optimization. A DIFNO is an FNO trained by minimizing its prediction error jointly on output and Fréchet derivative samples of a high-fidelity operator (e.g., a parametric PDE solution operator). As a result, a DIFNO can closely emulate not only the high-fidelity operator's response but also its sensitivities. To motivate the use of DIFNOs instead of conventional FNOs as surrogate models, we show that accurate surrogate-driven PDE-constrained optimization requires accurate surrogate Fréchet derivatives. Then, we establish (i) simultaneous universal approximation of continuously differentiable operators and their Fréchet derivatives by FNOs on compact sets, and (ii) universal approximation of continuously differentiable operators by FNOs in weighted Sobolev spaces with input measures that have unbounded supports. Our theoretical results certify the capability of FNOs for accurate derivative-informed operator learning and for the solution of PDE-constrained optimization problems. Furthermore, we develop efficient training schemes that leverage dimensionality reduction and multi-resolution techniques to significantly reduce memory and computational costs in Fréchet derivative learning. Numerical examples on nonlinear diffusion--reaction, Helmholtz, and Navier--Stokes equations demonstrate that DIFNOs are superior in sample complexity for operator learning and solving infinite-dimensional PDE-constrained inverse problems, achieving high accuracy at low training sample sizes.

cs.LG↗

arXiv2025

Learning Memory and Material Dependent Constitutive Laws

We propose and study a neural operator framework for learning memory- and material microstructure-dependent constitutive laws for heterogeneous materials. We work in the two-scale setting where homogenization theory provides a systematic approach to deriving macroscale constitutive laws, obviating the need to resolve complex microstructure repeatedly. However, the unit cell problems defining these constitutive models are typically not amenable to explicit evaluation. It is therefore of interest to learn constitutive models from data generated by the unit cell problem. Our proposed framework models homogenized constitutive laws with both memory- and microstructure-dependence through the use of Markovian recurrent and Fourier neural operators. The homogenization problem for Kelvin-Voigt viscoelastic materials is studied to provide firm theoretical foundations for our model. The theoretical properties of the cell problem in this Kelvin-Voigt setting motivate the proposed learning framework; and are also used to prove a universal approximation theorem for the learned macroscale constitutive model. Numerical experiments show that the proposed learning framework accurately learns memory- and microstructure-dependent viscoelastic and elasto-viscoplastic constitutive models, beyond the setting of the theory. Furthermore, we show that the learned constitutive models can be successfully deployed in macroscale simulation of material deformation for different microstructures without retraining.

math.NA↗

arXiv2024

LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport

We present LazyDINO, a transport map variational inference method for fast, scalable, and efficiently amortized solutions of high-dimensional nonlinear Bayesian inverse problems with expensive parameter-to-observable (PtO) maps. Our method consists of an offline phase in which we construct a derivative-informed neural surrogate of the PtO map using joint samples of the PtO map and its Jacobian. During the online phase, when given observational data, we seek rapid posterior approximation using surrogate-driven training of a lazy map [Brennan et al., NeurIPS, (2020)], i.e., a structure-exploiting transport map with low-dimensional nonlinearity. The trained lazy map then produces approximate posterior samples or density evaluations. Our surrogate construction is optimized for amortized Bayesian inversion using lazy map variational inference. We show that (i) the derivative-based reduced basis architecture [O'Leary-Roseberry et al., Comput. Methods Appl. Mech. Eng., 388 (2022)] minimizes the upper bound on the expected error in surrogate posterior approximation, and (ii) the derivative-informed training formulation [O'Leary-Roseberry et al., J. Comput. Phys., 496 (2024)] minimizes the expected error due to surrogate-driven transport map optimization. Our numerical results demonstrate that LazyDINO is highly efficient in cost amortization for Bayesian inversion. We observe one to two orders of magnitude reduction of offline cost for accurate posterior approximation, compared to simulation-based amortized inference via conditional transport and conventional surrogate-driven transport. In particular, LazyDINO outperforms Laplace approximation consistently using fewer than 1000 offline samples, while other amortized inference methods struggle and sometimes fail at 16,000 offline samples.

math.NA↗

arXiv2024

Derivative-informed neural operator acceleration of geometric MCMC for infinite-dimensional Bayesian inverse problems

We propose an operator learning approach to accelerate geometric Markov chain Monte Carlo (MCMC) for solving infinite-dimensional Bayesian inverse problems (BIPs). While geometric MCMC employs high-quality proposals that adapt to posterior local geometry, it requires repeated computations of gradients and Hessians of the log-likelihood, which becomes prohibitive when the parameter-to-observable (PtO) map is defined through expensive-to-solve parametric partial differential equations (PDEs). We consider a delayed-acceptance geometric MCMC method driven by a neural operator surrogate of the PtO map, where the proposal exploits fast surrogate predictions of the log-likelihood and, simultaneously, its gradient and Hessian. To achieve a substantial speedup, the surrogate must accurately approximate the PtO map and its Jacobian, which often demands a prohibitively large number of PtO map samples via conventional operator learning methods. In this work, we present an extension of derivative-informed operator learning [O'Leary-Roseberry et al., J. Comput. Phys., 496 (2024)] that uses joint samples of the PtO map and its Jacobian. This leads to derivative-informed neural operator (DINO) surrogates that accurately predict the observables and posterior local geometry at a significantly lower training cost than conventional methods. Cost and error analysis for reduced basis DINO surrogates are provided. Numerical studies demonstrate that DINO-driven MCMC generates effective posterior samples 3--9 times faster than geometric MCMC and 60--97 times faster than prior geometry-based MCMC. Furthermore, the training cost of DINO surrogates breaks even compared to geometric MCMC after just 10--25 effective posterior samples.

math.NA↗

arXiv2023

Bayesian model calibration for diblock copolymer thin film self-assembly using power spectrum of microscopy data and machine learning surrogate

Identifying parameters of computational models from experimental data, or model calibration, is fundamental for assessing and improving the predictability and reliability of computer simulations. In this work, we propose a method for Bayesian calibration of models that predict morphological patterns of diblock copolymer (Di-BCP) thin film self-assembly while accounting for various sources of uncertainties in pattern formation and data acquisition. This method extracts the azimuthally-averaged power spectrum (AAPS) of the top-down microscopy characterization of Di-BCP thin film patterns as summary statistics for Bayesian inference of model parameters via the pseudo-marginal method. We derive the analytical and approximate form of a conditional likelihood for the AAPS of image data. We demonstrate that AAPS-based image data reduction retains the mutual information, particularly on important length scales, between image data and model parameters while being relatively agnostic to the aleatoric uncertainties associated with the random long-range disorder of Di-BCP patterns. Additionally, we propose a phase-informed prior distribution for Bayesian model calibration. Furthermore, reducing image data to AAPS enables us to efficiently build surrogate models to accelerate the proposed Bayesian model calibration procedure. We present the formulation and training of two multi-layer perceptrons for approximating the parameter-to-spectrum map, which enables fast integrated likelihood evaluations. We validate the proposed Bayesian model calibration method through numerical examples, for which the neural network surrogate delivers a fivefold reduction of the number of model simulations performed for a single calibration task.

physics.comp-ph↗

arXiv2022

Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems

We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability to approximate the parameter-to-solution maps defined by PDEs using as training data solutions of PDEs at a limited number of parameter samples. The computational cost of BIPs can be drastically reduced if the large number of PDE solves required for posterior characterization are replaced with evaluations of trained neural operators. However, reducing error in the resulting BIP solutions via reducing the approximation error of the neural operators in training can be challenging and unreliable. We provide an a priori error bound result that implies certain BIPs can be ill-conditioned to the approximation error of neural operators, thus leading to inaccessible accuracy requirements in training. To reliably deploy neural operators in BIPs, we consider a strategy for enhancing the performance of neural operators, which is to correct the prediction of a trained neural operator by solving a linear variational problem based on the PDE residual. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error, all while retaining substantial computational speedups of posterior sampling when models are governed by highly nonlinear PDEs. The strategy is applied to two numerical examples of BIPs based on a nonlinear reaction--diffusion problem and deformation of hyperelastic materials. We demonstrate that posterior representations of the two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction.

math.NA↗

arXiv2022

Optimal design of chemoepitaxial guideposts for directed self-assembly of block copolymer systems using an inexact-Newton algorithm

Directed self-assembly (DSA) of block-copolymers (BCPs) is one of the most promising developments in the cost-effective production of nanoscale devices. The process makes use of the natural tendency for BCP mixtures to form nanoscale structures upon phase separation. The phase separation can be directed through the use of chemically patterned substrates to promote the formation of morphologies that are essential to the production of semiconductor devices. Moreover, the design of substrate pattern can formulated as an optimization problem for which we seek optimal substrate designs that effectively produce given target morphologies. In this paper, we adopt a phase field model given by a nonlocal Cahn--Hilliard partial differential equation (PDE) based on the minimization of the Ohta--Kawasaki free energy, and present an efficient PDE-constrained optimization framework for the optimal design problem. The design variables are the locations of circular- or strip-shaped guiding posts that are used to model the substrate chemical pattern. To solve the ensuing optimization problem, we propose a variant of an inexact Newton conjugate gradient algorithm tailored to this problem. We demonstrate the effectiveness of our computational strategy on numerical examples that span a range of target morphologies. Owing to our second-order optimizer and fast state solver, the numerical results demonstrate five orders of magnitude reduction in computational cost over previous work. The efficiency of our framework and the fast convergence of our optimization algorithm enable us to rapidly solve the optimal design problem in not only two, but also three spatial dimensions.

math.OC↗

arXiv2022

Bayesian model calibration for block copolymer self-assembly: Likelihood-free inference and expected information gain computation via measure transport

We consider the Bayesian calibration of models describing the phenomenon of block copolymer (BCP) self-assembly using image data produced by microscopy or X-ray scattering techniques. To account for the random long-range disorder in BCP equilibrium structures, we introduce auxiliary variables to represent this aleatory uncertainty. These variables, however, result in an integrated likelihood for high-dimensional image data that is generally intractable to evaluate. We tackle this challenging Bayesian inference problem using a likelihood-free approach based on measure transport together with the construction of summary statistics for the image data. We also show that expected information gains (EIGs) from the observed data about the model parameters can be computed with no significant additional cost. Lastly, we present a numerical case study based on the Ohta--Kawasaki model for diblock copolymer thin film self-assembly and top-down microscopy characterization. For calibration, we introduce several domain-specific energy- and Fourier-based summary statistics, and quantify their informativeness using EIG. We demonstrate the power of the proposed approach to study the effect of data corruptions and experimental designs on the calibration results.

physics.comp-ph↗

arXiv2020

A globally convergent modified Newton method for the direct minimization of the Ohta-Kawasaki energy with application to the directed self-assembly of diblock copolymers

We propose a fast and robust scheme for the direct minimization of the Ohta-Kawasaki energy that characterizes the microphase separation of diblock copolymer melts. The scheme employs a globally convergent modified Newton method with line search which is shown to be mass-conservative, energy-descending, asymptotically quadratically convergent, and three orders of magnitude more efficient than the commonly-used gradient flow approach. The regularity and the first-order condition of minimizers are analyzed. A numerical study of the chemical substrate guided directed self-assembly of diblock copolymer melts, based on a novel polymer-substrate interaction model and the proposed scheme, is provided.

physics.comp-ph↗