Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects
Zhentao Tan, Yuze Hao, Boyi Zou, Mingsheng Long, Yi Yang, Gang Bao
Abstract
Solving inverse partial differential equation (PDE) problems is a fundamental topic in scientific research due to its broad significance across a wide range of real-world applications. Inverse PDE problems arise across medical imaging, geophysics, materials science, and aerodynamics, where the goal is to infer hidden causes, design structures, or control physical states. In this paper, we provide a comprehensive review of recent advances in solving inverse PDE problems using artificial intelligence (AI). We first introduce the basic formulation, key challenges, and traditional numerical foundations of inverse PDE problems, and then organize it into three major categories: inverse problems, inverse design, and control problems. For each category, we further present a methodological paradigms, and review representative state-of-the-art approaches from recent years. We then summarize representative applications across scientific and industrial domains, including mechanical systems, aerodynamic problems, thermal systems, full-waveform inversion, system identification, and medical imaging. Finally, we discuss open challenges and future prospects, such as physics-informed architectures, limited real-world data, uncertainty quantification, and inverse foundation models. This survey aims to provide the first unified and systematic perspective on AI for inverse PDE problems, demonstrating how modern learning-based methods are reshaping inverse problems, inverse design, and control problems in PDE-governed systems.
AI Impact Assessments
(1 models)Scientific Impact Assessment
Core Contribution
This paper presents a comprehensive survey of AI-based methods for solving inverse partial differential equation (PDE) problems. Its primary contribution is a unifying taxonomy that organizes the rapidly expanding literature into three major categories—inverse problems, inverse design, and control problems—and further decomposes each into methodological paradigms (e.g., data-driven, optimization-based, generative, reinforcement learning). The paper claims to be "the first unified and systematic perspective on AI for inverse PDE problems," distinguishing itself from prior surveys (e.g., [220, 259]) that focused primarily on forward PDE simulation.
The taxonomy is well-structured: inverse problems are subdivided into inverse-mapping, PDE-constrained, surrogate-based, inverse dynamics, and generative methods; inverse design into optimization-based, direct, and generative approaches; and control problems into optimization-based, offline-trained, and reinforcement learning methods. Each category includes detailed formulations and comprehensive tables of representative methods with their venues, backbones, and application domains.
Methodological Rigor
As a survey, the paper's rigor lies in its organizational framework and literature coverage rather than experimental validation. The taxonomy is logically coherent and mathematically grounded—each category is introduced with formal problem formulations (Equations 5, 13–14, 15), making the distinctions between inverse problems, inverse design, and control problems precise. The paper covers approximately 270+ references spanning major venues (NeurIPS, ICML, ICLR, Nature, Science, domain-specific journals).
However, there are notable gaps in rigor. The paper lacks quantitative comparisons or meta-analyses of methods across categories. There are no benchmark tables comparing methods on shared datasets, which would have strengthened the survey's utility as a practical reference. The distinction between some categories (e.g., surrogate-based inverse problems vs. learned-model-assisted inverse design) could be sharpened further—the boundaries sometimes feel more organizational than fundamental. Additionally, while the paper mentions challenges like ill-posedness extensively, it doesn't provide a systematic analysis of which methods are better suited for specific types of ill-posedness.
Potential Impact
The survey has significant potential impact for multiple communities:
1. Bridge-building: By unifying inverse problems, inverse design, and control under a single framework, the paper could facilitate cross-pollination between communities that have historically developed methods independently (e.g., computational imaging, topology optimization, fluid control).
2. Entry point for researchers: The comprehensive tables (Tables 1–3) and taxonomic figures (Figs. 1–5) provide an efficient entry point for researchers new to AI-for-inverse-PDEs, potentially lowering barriers to entry.
3. Future directions: The discussion of open challenges—physics-informed architectures, uncertainty quantification, inverse foundation models, and limited real-world data—identifies genuinely important research directions. The concept of "inverse foundation models" is particularly timely and could catalyze new research.
4. Application breadth: Covering six application domains (mechanical systems, aerodynamics, thermal systems, FWI, system identification, medical imaging) demonstrates the cross-disciplinary relevance.
Timeliness & Relevance
The survey is highly timely. The intersection of AI and inverse PDE problems has seen explosive growth, with many key papers appearing in 2023–2025 (DiffusionPDE, FunDPS, DiffPhyCon, etc.). There is a genuine need for a unifying perspective as methods from diffusion models, neural operators, and reinforcement learning converge on inverse PDE problems from different angles. The discussion of foundation models for inverse problems is particularly relevant given the current trajectory of the field toward general-purpose scientific AI systems.
Strengths
1. Comprehensive coverage: The paper covers an impressive breadth of methods and applications, with well-organized tables that serve as practical reference guides.
2. Clear taxonomy: The three-way decomposition (inverse problems / inverse design / control) with further methodological subdivision is intuitive and well-motivated by the distinct challenges each category presents.
3. Formal grounding: Each category is introduced with proper mathematical formulations, making the survey accessible to both ML and applied mathematics audiences.
4. Problem-specific considerations: The dedicated discussions on design space representation (Section 4.4) and temporal/feedback structure (Section 5.4) add depth beyond a simple literature enumeration.
5. Forward-looking discussion: The challenges section identifies substantive open problems rather than generic platitudes.
Limitations
1. No quantitative benchmarking: The survey would be substantially more impactful with comparative tables showing method performance on shared benchmarks (e.g., PDEBench tasks). Without this, practitioners cannot easily select methods for their use cases.
2. Limited critical analysis: The paper tends toward cataloging rather than critically evaluating methods. For instance, when do PINNs fundamentally fail for inverse problems? When should one prefer diffusion-based methods over surrogate-based optimization? These practical decision-making insights are largely absent.
3. Incomplete coverage of classical methods: Section 2.3 on traditional methods is brief. A survey claiming "Past, Present, and Prospects" should provide stronger historical context to properly situate AI contributions against classical baselines.
4. Scalability and computational cost: The survey rarely discusses computational costs, training data requirements, or scalability properties of the reviewed methods—critical practical considerations.
5. Reproducibility: No discussion of code availability, standardized benchmarks, or reproducibility practices across the surveyed methods.
6. Some organizational overlap: The boundary between "inverse problems" and "inverse design" is sometimes blurred in practice, and the paper could more explicitly discuss when methods from one category apply to another.
Overall Assessment
This is a well-organized and timely survey that fills a genuine gap in the literature by providing the first unified treatment of AI methods across inverse problems, inverse design, and control for PDE-governed systems. Its main value lies in its comprehensive taxonomy and extensive literature coverage. However, it falls short of being a definitive reference due to the absence of quantitative comparisons, limited critical evaluation of method trade-offs, and insufficient discussion of practical considerations like computational costs and method selection guidelines. The paper will likely serve as a useful starting point for researchers entering this field, though its long-term citation impact may be moderate given the rapid pace of the field.
Generated May 19, 2026
Comparison History (19)
Paper 2 likely has higher scientific impact due to its broad, cross-disciplinary scope (inverse problems, design, and control across many PDE-governed domains) and timeliness as a unifying AI survey that can shape research agendas, standardize terminology, and influence many fields. Paper 1 is novel and application-relevant, but is narrower (nanomedicine-specific) and its reported performance suggests moderate effectiveness and reliance on expert judgment, which may limit immediate uptake. Overall, Paper 2’s breadth and potential to become a widely cited reference give it higher impact potential.
Paper 1 provides a comprehensive, first-of-its-kind survey unifying AI approaches for inverse PDE problems across three major categories (inverse problems, inverse design, and control), covering broad applications from medical imaging to aerodynamics. Its scope, systematic organization, and coverage of open challenges position it as a foundational reference for a large and growing research community. Paper 2, while introducing a valuable benchmark for LLM-based scientific assistants, addresses a narrower problem (multi-turn clarification) with more limited immediate applicability and a smaller target audience.
Paper 1 is more likely to have higher impact due to its novel, testable contributions (a small chess LM outperforming larger baselines, a brittleness/generalization critique, and a verifier-in-the-loop method with large gains), strong methodological emphasis on controlled evaluation, and open-sourced code/data/checkpoints enabling immediate follow-on work. Its verifier+LLM framing generalizes to other well-defined symbolic domains beyond chess, aligning with timely concerns about LLM memorization vs. reasoning. Paper 2 is a valuable, broad survey with wide applicability, but as a review it is less methodologically innovative and typically yields lower scientific impact than a reusable new method/dataset.
Paper 2 has higher potential impact because it introduces a concrete, timely methodological innovation—RAG+LLM for sentence-level EEG-to-text decoding—validated on real EEG data with a strict no-teacher-forcing inference protocol and statistical significance. If it generalizes, it could advance practical BCI communication, a high-impact real-world application, and influence multiple areas (neuroscience, ML, NLP, assistive tech). Paper 1 is a broad survey; while useful and wide-ranging, reviews typically contribute less direct scientific novelty and immediate downstream methodological change than a new validated pipeline.
Paper 1 likely has higher impact due to clear technical novelty (trainable binary masks for token-adaptive MoE routing), demonstrated large practical gains (up to 85% FLOPs reduction, 2.5× decoding speedup) and deployment-oriented engineering (custom CUDA kernel, vLLM integration). It is timely given widespread MoE adoption and inference-cost pressure, and can be broadly adopted across MoE-based LLM systems. Paper 2 is a valuable survey with broad relevance, but surveys typically contribute less direct methodological innovation and measurable performance impact than a deployable new technique.
Paper 1 offers a comprehensive and unified review of AI applied to inverse PDE problems, a crucial area bridging machine learning with physics, engineering, and medicine. Its broad applicability to real-world challenges like medical imaging and aerodynamics gives it a wider cross-disciplinary impact compared to Paper 2, which focuses on a specific spatial and temporal reasoning benchmark for language models. The foundational nature and expansive scope of Paper 1 indicate a higher potential to shape future scientific research.
Paper 1 is a comprehensive survey covering AI for inverse PDE problems—a fundamental topic with broad applications across medicine, geophysics, materials science, and aerodynamics. Its scope spans three major problem categories with a unified taxonomy, addressing a wide audience across multiple scientific and engineering fields. Paper 2 addresses a more niche problem (cross-system medical knowledge alignment between TCM and Western medicine) with narrower applicability. The breadth, timeliness, and cross-disciplinary relevance of Paper 1 give it substantially higher potential for scientific impact and citation.
Paper 1 provides the first unified survey of AI for inverse PDE problems, covering a broad and fundamental topic with applications across medical imaging, geophysics, materials science, and aerodynamics. Its comprehensive taxonomy and systematic review serve as a foundational reference for a large research community. Paper 2 introduces an interesting benchmark for evaluating AI cognitive alignment, but its scope is narrower and more niche. Paper 1's breadth of impact across multiple scientific and engineering fields, combined with the growing importance of AI-driven scientific computing, gives it substantially higher potential for citations and influence.
Paper 2 addresses a fundamental and mathematically profound topic (inverse PDE problems) with broad applications across physics, medical imaging, and engineering. As a comprehensive review in the rapidly growing field of AI for Science, it is likely to garner high citation counts and influence multiple disciplines. Paper 1, while useful, addresses a highly specific niche (Scrum Masters in agile meetings) using existing AI tools, resulting in a much narrower potential scientific impact.
Paper 1 is more novel and timely: it introduces a new benchmark/framework (EnvTrustBench) targeting a specific, under-evaluated reliability failure mode in LLM agents (evidence-grounding defects) with direct security implications. It includes an executable methodology, defined defect taxonomy, and empirical evaluation across multiple models/scaffolds, enabling reproducible comparisons and follow-on work. Paper 2 is a broad survey; while potentially influential as a reference, it is less methodologically innovative and mainly synthesizes existing results. Paper 1’s tool-like artifact and relevance to rapidly deployed agent systems suggest higher near-term scientific and practical impact.
Paper 2 introduces a highly novel methodological innovation (Latent Action Reparameterization) addressing a critical bottleneck in a rapidly growing field: LLM agent inference efficiency. Its original contribution to action representation learning offers direct, scalable improvements to AI systems. In contrast, while Paper 1 covers a broad and impactful interdisciplinary domain, it is a review article summarizing existing work rather than introducing new methodological breakthroughs, giving Paper 2 a higher potential for direct scientific advancement.
Paper 2 likely has higher scientific impact due to its broad, timely scope and cross-domain relevance: AI methods for inverse PDEs affect many high-impact areas (imaging, geophysics, engineering, materials) and are currently a fast-moving research frontier. As a unifying survey, it can shape agendas, standardize terminology, and accelerate adoption across fields. Paper 1 is novel and potentially important for Semantic Web/knowledge graphs, but its impact is more specialized and depends on successful standardization/uptake of proposed frameworks, whereas inverse-PDE AI is already central to multiple communities.
Paper 1 presents a novel, applied AI architecture that directly accelerates experimental workflows across multiple disciplines (chemistry, biology, materials science) by automating laboratory protocols. While Paper 2 is a valuable and comprehensive survey on AI for inverse PDEs, Paper 1 offers a tangible, highly innovative tool with immediate, real-world utility that fundamentally changes how scientists interact with automated labs, representing a more significant leap in original research and practical scientific impact.
Paper 1 introduces a novel, highly timely benchmark addressing a critical vulnerability in modern AI: academic integrity and data fabrication by AI systems. As AI becomes increasingly autonomous in research, exposing and mitigating its intrinsic completion bias is crucial across all scientific disciplines. While Paper 2 is a valuable and likely highly-cited survey on AI for inverse PDEs, Paper 1 represents original research that defines a new evaluation paradigm, likely sparking a new subfield of AI safety in scientific discovery and driving immediate methodological improvements in LLM training.
Paper 2 is a comprehensive survey covering AI for inverse PDE problems across multiple scientific domains (medical imaging, geophysics, materials science, aerodynamics). Surveys that provide unified taxonomies for rapidly growing fields tend to have high citation impact as reference works. Its breadth spans numerous disciplines, and inverse PDEs are fundamental to many scientific applications. Paper 1, while addressing an important safety-critical niche (clinical acuity benchmarking for LLMs), has a narrower scope focused on medical triage evaluation. Paper 2's broader cross-disciplinary relevance and foundational nature give it higher potential impact.
Paper 1 is a comprehensive survey covering AI for inverse PDE problems—a fundamental topic spanning medical imaging, geophysics, materials science, and aerodynamics. Its breadth, systematic taxonomy, and identification of open challenges make it a high-impact reference work for a large research community. Paper 2 presents a novel but narrower contribution (latent-space optimization for heuristic design) with results described as 'competitive with' existing baselines rather than clearly superior, limiting its immediate impact. The survey's cross-disciplinary relevance and timeliness give it substantially broader potential influence.
Paper 1 proposes a highly novel, original methodological advancement by integrating physiological ODE priors into latent diffusion models for clinical simulation. While Paper 2 is a valuable survey on AI for PDEs, Paper 1 introduces a concrete, innovative solution to a critical real-world problem (intervention-aware clinical decision support). Its rigorous approach to handling uncertainty and mitigating generative hallucinations in a high-stakes medical context demonstrates greater potential for driving immediate, transformative applied impact in healthcare AI.
Paper 2 presents a novel, concrete contribution—identifying 'Entropy-Gradient Inversion' as a mechanistic fingerprint of reasoning in LRMs and proposing CorR-PO, a new RL optimization method that demonstrably outperforms baselines. This offers both theoretical insight into LRM internals and a practical training method, addressing a timely and high-impact problem in AI reasoning. Paper 1, while comprehensive and useful as a survey of AI for inverse PDE problems, synthesizes existing work rather than introducing new methods. Original contributions with validated empirical results typically have higher citation and adoption impact than review papers.
Paper 2 is a comprehensive survey covering AI for inverse PDE problems—a foundational topic spanning medical imaging, geophysics, materials science, and aerodynamics. Its breadth of impact across multiple scientific and engineering fields, combined with its role as the first unified systematic review of this area, gives it high citation potential and broad relevance. Paper 1, while methodologically interesting, addresses a narrow domain (a single cybersecurity benchmark) with explicitly acknowledged limited generalizability, restricting its broader scientific impact.