adjoint-state method

This study addresses the lack of fair comparison between adjoint-based optimization and physics-informed neural networks (PINNs) in solving PDE-constrained inverse problems by conducting a systematic evaluation within a unified framework, ensuring identical settings for governing equations, observation models, regularization, parameterization, and optimizers. The analysis reveals that the representation of unknown fields critically determines method performance: grid-discretized fields favor adjoint methods, whereas neural network representations align better with PINNs. Building on this insight, the authors propose a hybrid strategy that uses PINN-based pre-training to initialize adjoint optimization, achieving high accuracy while substantially reducing computational cost. This work presents the first equitable benchmark between these two dominant approaches and establishes a new paradigm for efficiently solving high-dimensional spatiotemporal inverse problems.

This work proposes a novel framework that integrates continuous-time operator inference with the adjoint-state method to address the poor accuracy and unstable extrapolation of traditional data-driven reduced-order models under sparse sampling and noisy data. By minimizing trajectory loss during training, the approach avoids direct differentiation of noisy measurements and leverages temporal integration for intrinsic regularization. For the first time, the adjoint method is incorporated into continuous-time operator inference, enabling efficient gradient computation and stable optimization. Combining continuous adjoint equations, projected snapshot matching, and gradient-based optimization, the method demonstrates significantly improved accuracy and rolling prediction stability over standard operator inference when tested on the Burgers, Fisher–KPP, and convection–diffusion equations under sparse or noisy data conditions.

Data-driven discovery of partial differential equations (PDEs) remains challenging due to structural ambiguity and sensitivity to noise and data scale. Method: We propose an adjoint-based parametric modeling framework for PDE discovery. A sparse candidate library—comprising linear/nonlinear terms and spatial derivatives—is used to parameterize the PDE form, yielding a PDE-constrained optimization problem. For the first time, we systematically derive the adjoint equations for general parametric PDE families via variational calculus, enabling machine-precision analytical gradient computation. Contribution/Results: Our method significantly outperforms sparse regression approaches (e.g., PDE-FIND) in structural identification accuracy and noise robustness across diverse PDEs—including Burgers, KdV, and reaction-diffusion equations—especially under high noise levels and large-scale data. Integrated forward and adjoint numerical solvers ensure efficient training, while analytical gradients substantially accelerate optimization convergence.

In dynamic optimization, conventional adjoint methods require storing the full spatiotemporal wavefield, resulting in memory consumption scaling linearly with problem size—severely limiting scalability for large-scale problems. To address this, we propose an approximate adjoint method grounded in the superposition principle, reducing memory complexity for sensitivity computation from *O(TN)* to *O(N)*, where *T* is the number of time steps and *N* the number of spatial degrees of freedom. The method avoids storing the entire time-history wavefield, instead retaining only a few localized temporal states. Integrated with a CUDA-accelerated finite-difference forward solver, it enables iterative sensitivity updates. On an NVIDIA A100 GPU, we achieve, for the first time, billion-parameter-scale dynamic full-waveform inversion and transient acoustic topology optimization. Memory usage is reduced by one to two orders of magnitude, with controlled accuracy degradation (<5%).

This study addresses the simultaneous inverse reconstruction of multiple physical coefficients in partial differential equations (PDEs). We propose a model-consistent, data-driven iterative reconstruction framework. Methodologically, we first integrate physics-informed neural networks (PINNs), PDE model constraints, and uncertainty quantification in a unified manner—enabling joint assimilation of observational data and prior knowledge about coefficients, while explicitly modeling how learning uncertainty propagates into inversion outcomes during iteration. Our key contribution lies in establishing an intrinsic consistency mechanism between data-driven modeling and PDE-based physical constraints. Numerical experiments on two representative inverse problems demonstrate significant improvements in multi-coefficient joint reconstruction accuracy, alongside enhanced robustness and cross-scenario generalization capability.

This work addresses the computational inefficiency of existing methods for solving inverse problems with diffusion priors under nonlinear forward operators, which often rely on expensive repeated derivative evaluations or inner-loop optimization/MCMC sampling. The authors propose a training-free solver that replaces inner loops with hard constraint projections in measurement space and analytically derived optimal step sizes, yielding fixed and low per-noise-level computational cost. The key innovation lies in the first joint optimization framework that is adjoint-free, combining analytical step sizes with hard constraints. This approach integrates ADMM splitting, reannealing, and a hybrid latent/pixel-space scheduling strategy to guarantee local optimality and descent properties, while also enabling a derived KL error bound. Experiments demonstrate state-of-the-art PSNR, SSIM, and LPIPS performance in image reconstruction, achieving up to 19.5× acceleration without requiring hand-coded adjoints or MCMC sampling.

This work addresses the ill-posedness of inverse problems governed by partial differential equations, which arises from data noise, missing observations, and non-uniqueness, and for which existing Bayesian methods struggle to enforce hard physical constraints effectively. The authors propose a dual-space sampling framework that uniquely integrates the augmented Lagrangian method, the alternating direction method of multipliers (ADMM), and Stein variational gradient descent (SVGD) to transform hard constraints into differentiable penalty terms. This approach enables efficient posterior sampling while strictly satisfying physical laws. It combines the well-conditioned nature of dual solvers with the nonparametric expressiveness of SVGD. Experiments on Rosenbrock inference, Gaussian anomaly modeling, and Marmousi II full-waveform inversion demonstrate that the method yields well-calibrated uncertainty estimates, with posterior distributions converging stably as data coverage increases.

This work addresses the lack of systematic benchmarks and evaluation frameworks for inverse problems involving partial differential equations (PDEs) by introducing PDEInvBench, the first comprehensive benchmark dataset encompassing numerical simulations of diverse time-varying and time-invariant PDEs. The study systematically explores the neural network design space across three dimensions—optimization strategies, problem representations, and model/data scale—and evaluates architectures with varying inductive biases and conditioning strategies through supervised learning, self-supervised learning, and test-time optimization. Key findings reveal that a two-stage training protocol substantially enhances performance, incorporating PDE derivative features as inputs consistently improves accuracy, and increasing the diversity of initial conditions yields greater gains than merely expanding the parameter range.

This work addresses the challenges of heavy reliance on large paired supervised datasets and posterior sampling over-smoothing in inverse problems for partial differential equations (PDEs). To this end, we propose the Decoupled Diffusion Inverse Solver (DDIS), which leverages an unconditional diffusion model to learn the prior over coefficients and explicitly models the forward PDE using a neural operator, thereby enabling physics-informed, efficient inversion. Our method introduces a decoupled architecture that mitigates guidance decay under data scarcity by avoiding joint modeling, and incorporates a novel Decoupled Annealed Posterior Sampling (DAPS) strategy to effectively suppress over-smoothing. Experiments demonstrate that with only 1% of the training data, DDIS reduces the L2 error by 40% compared to joint models; in sparse observation settings, it achieves average reductions of 11% in L2 error and 54% in spectral error.

This work addresses the challenge of efficiently solving high-dimensional parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The authors propose a novel reduced-order modeling framework that combines finite element discretization in the physical domain with surrogate modeling in the parameter space: classical interpolation is employed for low-dimensional parameters, while extreme learning machines (ELMs) are introduced for the first time to handle high-dimensional settings. The method preserves Sobolev regularity and offers theoretically guaranteed error bounds, significantly enhancing computational efficiency. Moreover, it provides rigorous reconstruction error estimates for inverse problems. Demonstrated in quantitative photoacoustic tomography, the approach achieves high-fidelity recovery of both parameters and potential functions at a fraction of the computational cost of conventional methods, all while maintaining provable error guarantees.