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.

翻译：受偏微分方程（PDE）约束的反问题是计算力学的核心问题，通常通过伴随方法进行优化求解，而物理信息神经网络（PINNs）作为一种灵活的替代方案逐渐兴起。由于这两种方法通常在不同的公式化、参数化、优化器和正则化选择下进行比较，其相对性能难以评估。我们针对PDE约束反问题，对伴随优化和PINNs进行了公平的比较。基于共同的抽象公式，我们在相同的域、控制方程、观测模型和正则化项上实例化两种方法，并在适用情况下匹配优化器、未知参数化和算术精度。基准测试包括非定常Burgers方程、含噪声的Darcy渗透率反演、三维Allen-Cahn反应项识别以及非定常Navier-Stokes粘性识别。结果表明，未知量的表示很大程度上决定了优选方法：基于网格的场更适合离散伴随方法，而神经网络表示则是PINNs的原生形式，适用于封闭模型和本构模型。对于时间相关问题，伴随反演常因轨迹存储和微分计算而开销巨大，而PINNs能以较低成本提供满意的重建结果。采用PINNs热启动的伴随策略可大幅降低计算成本，同时恢复伴随级别的精度。