Traditional and Machine Learning Approaches to Partial Differential Equations: A Critical Review of Methods, Trade-Offs, and Integration

The solution of partial differential equations (PDEs) underpins computational modeling across science and engineering, from quantum mechanics to climate dynamics. This review examines the current landscape of PDE solving methods, encompassing both traditional numerical approaches that have been refined over decades and emerging machine learning techniques that fundamentally transform computational paradigms. We systematically analyze classical methods, evaluating their mathematical foundations, computational characteristics, and fundamental limitations. The review then explores eleven major families of machine learning-based approaches: physics-informed neural networks, neural operators, graph neural networks, transformer architectures, generative models, hybrid methods, meta-learning frameworks, physics-enhanced deep surrogates, random feature methods, DeePoly framework, and specialized architectures. Through detailed comparative analysis of over 40 distinct ML methods, we assess performance across multiple dimensions, including computational complexity, accuracy bounds, multiscale capability, uncertainty quantification, and implementation requirements. Our critical evaluation reveals fundamental trade-offs: traditional methods excel in providing provable accuracy guarantees and rigorous error bounds, but face scaling challenges for high-dimensional problems; neural methods, on the other hand, provide unprecedented computational speed and flexibility, but lack rigorous error control and systematic uncertainty quantification. Hybrid approaches that combine physical constraints with learning capabilities are increasingly dominating the accuracy-efficiency frontier. We identify key challenges, including the absence of comprehensive theoretical frameworks for neural methods, limited software maturity, and the need for verification requirements in safety-critical applications. The review concludes by outlining future directions, including foundation models that demonstrate zero-shot generalization across PDE families, as well as quantum and neuromorphic computing paradigms, and automated scientific discovery. Rather than replacement, we advocate for synthesis: leveraging complementary strengths of traditional and machine learning methods to expand the frontier of computationally tractable problems. This integration promises to enable breakthrough capabilities in understanding and controlling complex physical systems, from molecular to planetary scales.