做计算力学和逆问题研究的团队，这篇论文帮你省去选择方法的纠结——它给出了伴随法和PINN在不同场景下的明确优劣边界，看完可以直接指导你的实验设计。
该研究对偏微分方程约束逆问题中的伴随优化方法和物理信息神经网络进行了公平对比。通过统一抽象公式、匹配优化器、参数化及精度，在多个基准测试（如非稳态Burgers方程、噪声Darcy渗透率反演、三维Allen-Cahn反应识别、非稳态Navier-Stokes粘度识别）中评估两者性能。结果表明，未知参数的表示形式决定方法优劣：网格基场适合离散伴随，而神经表示是PINN的天然优势。对于时间依赖问题，伴随方法受轨迹存储和微分成本制约，而PINN能以更低成本获得满意重建。PINN热启动伴随策略能以大幅降低的成本恢复伴随级精度。
Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems
Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to assess because the two approaches are often compared under different formulations, parameterizations, optimizers, and regularization choices. We present a fair comparison of adjoint optimization and PINNs for PDE-constrained inverse problems. From a common abstract formulation, we instantiate both methods on identical domains, governing equations, observation models, and regularization terms, while matching the optimizer, unknown parameterization, and arithmetic precision wherever applicable. The benchmarks include unsteady Burgers, noisy Darcy permeability inversion, three-dimensional Allen--Cahn reaction identification, and unsteady Navier--Stokes viscosity identification. The results show that the representation of the unknown largely determines the preferred method: grid-based fields favor the discrete adjoint, whereas neural representations are native to PINNs and relevant for closure and constitutive modeling. For time-dependent problems, adjoint inversion can be dominated by trajectory storage and differentiation, while PINNs provide satisfactory reconstructions at lower cost. A PINN-warm-started adjoint strategy then recovers adjoint-level accuracy at substantially reduced cost.