Discovery, Understanding and Trust
Executive Statement
International Council for Industrial and Applied Mathematics (ICIAM), September 2026
A moment of extraordinary opportunity, and responsibility
Artificial intelligence is rapidly changing mathematical research, scientific computing and engineering. ICIAM welcomes these developments and believes that applied and industrial mathematics must have a strong voice in shaping them. AI can search enormous spaces, generate proofs and counterexamples, accelerate simulation and support scientific discovery. But mathematical progress is more than the production of answers, and scientific usefulness requires more than impressive benchmark performance. Understanding, validation, reliability, attribution and human judgement remain essential. In industry, AI is increasingly being combined with mathematical models and simulation to enable faster design, real-time prediction, digital twins and decision support. For ICIAM, the central opportunity is therefore a partnership in which AI extends human capabilities while mathematics provides structure, insight and trust.
1. AI can accelerate discovery, but discovery is more than producing answers
Proofs, counterexamples and failed approaches are all part of mathematical progress. A counterexample does not end a question: it asks us why a conjecture fails, which assumptions are missing, and what stronger or more meaningful formulation should replace it. The same principle should apply to AI-assisted mathematics. Knowing where an AI system failed, which approaches it explored unsuccessfully and under which conditions its reasoning broke down can be as informative as knowing where it succeeded. Selective reporting of successes alone gives an incomplete picture. Openness about failures, limitations and unsuccessful avenues will strengthen understanding, reproducibility and responsible evaluation.
2. Mathematics is essential for making AI trustworthy
When AI and machine learning become part of scientific and engineering computation, reliability is not optional. Applied mathematics provides tools for stability, robustness, error control, uncertainty quantification, physical consistency, causality and validation. These properties are particularly important when AI-generated predictions are embedded in engineering workflows, where an apparently accurate model may nevertheless lead to unstable simulations, non-physical behaviour or unreliable decisions. Incorporating known mathematical structure, such as conservation laws, symmetries and positivity properties, can reduce the learning problem, improve data efficiency and provide stronger guarantees. Mathematical analysis can help establish which properties are essential and how they can be preserved. Mathematics can help turn powerful data-driven models into scientific instruments whose behaviour and limitations can be understood and trusted.
3. Mathematics provides a powerful toolbox for AI
Decades of computational mathematics have produced highly efficient methods for differential equations, optimisation, inverse problems, multiscale computation, uncertainty quantification, model reduction and large-scale simulation. These methods are often fast, robust, interpretable and supported by mathematical analysis. They provide a powerful foundation for developing AI approaches that are efficient, reliable and interpretable. Indeed, many AI surrogate approaches echo ideas long used in model order reduction: substantial offline computation is used to construct a low-dimensional or learned representation that can then be evaluated rapidly online. Combining these established mathematical ideas with the capabilities of AI creates opportunities for methods that neither approach could achieve alone.
4. The future is a partnership of human insight, mathematics, data and AI
The interaction between mathematics and AI creates major new opportunities for science, engineering and industry. Mathematical knowledge can guide learning, reduce complexity and data requirements, improve reliability, and help determine when predictions can be trusted. AI, in turn, brings powerful new capabilities for search, approximation, synthesis and the exploitation of data. For industry, combining mathematical models, simulation, data and AI can make reliable mathematical prediction an integrated component of design, control and decision-making. Human insight determines which questions matter and how results should be interpreted; mathematics provides structure, models and constraints; data connect models to observations; AI provides new computational capabilities. The age of AI creates not less need for applied mathematics, but more.
5. The mathematical community must help shape the rules and infrastructure
ICIAM and its member societies have a responsibility to engage actively. Applied and industrial mathematicians have a particularly important role in building these bridges, because they work at the interfaces between mathematical models, numerical methods, data, scientific computing and real-world applications. AI-assisted mathematics and science require reliable verification, careful attribution, reproducibility and transparent provenance. The community should help develop open and robust mathematical validation environments, encourage disclosure of computational methods and resources, and create opportunities to learn from both successes and failures. We should also consider equitable access to the AI and computational infrastructure increasingly required for frontier research. Finally, we must educate mathematicians not merely to use AI tools, but to understand, analyse and improve them, and to recognise when established mathematical methods, AI approaches or a combination of both provide the best solution.
Our message
The appropriate response to AI is neither resistance nor uncritical enthusiasm, but active engagement. Mathematicians should use these tools, study them, improve them and help shape the conditions under which they are used. We should be ambitious about what human and artificial intelligence can achieve together, while remaining rigorous about correctness, transparency, reproducibility and scientific meaning.
AI can accelerate discovery. Mathematics can provide understanding and trust.
Together, human and artificial intelligence can open new directions for mathematics, science, engineering, industry and society.
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