Agentic Exploration of PDE Spaces using Latent Foundation Models for Parameterized Simulations

Abhijeet Vishwasrao; Francisco Giral; Mahmoud Golestanian; Federica Tonti; Andrea Arroyo Ramo; Adrian Lozano-Duran; Steven L. Brunton; Sergio Hoyas

cs.AIFeb 26, 2026

Agentic Exploration of PDE Spaces using Latent Foundation Models for Parameterized Simulations

Authors: Abhijeet Vishwasrao, Francisco Giral, Mahmoud Golestanian, Federica Tonti, Andrea Arroyo Ramo, Adrian Lozano-Duran, Steven L. Brunton, Sergio Hoyas,

Organizations: University of Michigan · Universidad Polit´ecnica de Madrid · Purdue University · Universitat Polit`ecnica de Val`encia ·

Abstract

Flow physics and more broadly physical phenomena governed by partial differential equations (PDEs), are inherently continuous, high-dimensional and often chaotic in nature. Traditionally, researchers have explored these rich spatiotemporal PDE solution spaces using laboratory experiments and/or computationally expensive numerical simulations. This severely limits automated and large-scale exploration, unlike domains such as drug discovery or materials science, where discrete, tokenizable representations naturally interface with large language models. We address this by coupling multi-agent LLMs with latent foundation models (LFMs), a generative model over parametrised simulations, that learns explicit, compact and disentangled latent representations of flow fields, enabling continuous exploration across governing PDE parameters and boundary conditions. The LFM serves as an on-demand surrogate simulator, allowing agents to query arbitrary parameter configurations at negligible cost. A hierarchical agent architecture orchestrates exploration through a closed loop of hypothesis, experimentation, analysis and verification, with a tool-modular interface requiring no user support. Applied to flow past tandem cylinders at Re = 500, the framework autonomously evaluates over 1,600 parameter-location pairs and discovers divergent scaling laws: a regime-dependent two-mode structure for minimum displacement thickness and a robust linear scaling for maximum momentum thickness, with both landscapes exhibiting a dual-extrema structure that emerges at the near-wake to co-shedding regime transition. The coupling of the learned physical representations with agentic reasoning establishes a general paradigm for automated scientific discovery in PDE-governed systems.

Figures & tables

Figure 1: Overview of the multi-agent exploration framework. A user-specified task initiates the exploration loop: the Planner formulates sampling strategies over the parameter space $\mathcal{P}$ , the Analyst executes these through a modular tool interface that routes physics queries to the latent foundation model (LFM) for on-demand flow-field generation and the Critic validates outputs before updating the shared knowledge base. This closed loop of strategy, experimentation and verification continues until convergence criteria are met, after which the Writer synthesizes findings into a structured report.

$\mathcal{S}/D$	$x_{\delta^{}}^{}/D$	$x_{\theta}^{*}/D$	$\|\Delta x^{*}\|/D$
3.5	0.50	2.15	1.65
4.5	0.50	3.05	2.55
5.5	0.50	4.10	3.60
6.5	0.65	5.45	4.80
7.5	2.75	6.35	3.60
8.5	3.65	7.25	3.60

Table 1: Extremal locations for minimum displacement thickness (

x_{\delta^{*}}^{*}

) and maximum momentum thickness (

x_{\theta}^{*}

) as discovered by the agent via LFM. A horizontal rule separates the near-wake and co-shedding regimes. The divergence

|\Delta x^{*}|

grows steadily in the near-wake regime, drops sharply at the regime transition (

\mathcal{S}\approx 7D

), then plateaus near

3.5\text{--}3.6D

in the co-shedding regime.

Figure 2: Extremal locations $x_{\delta^{*}}^{*}$ (blue circles, minimum $\delta^{*}$ ) and $x_{\theta}^{*}$ (orange squares, maximum $\theta$ ) as functions of cylinder spacing $\mathcal{S}$ . Filled markers: LFM-derived values; open markers: simulation ground truth. The minimum $\delta^{*}$ location transitions from a near-constant value ( $\approx 0.5D$ ) in the near-wake regime to a linearly growing trend ( $0.71\mathcal{S}-2.2D$ ) in the co-shedding regime. The maximum $\theta$ location follows a single linear law ( $\mathcal{S}-1.2D$ , $R^{2}=0.997$ ) across all spacings. The non-monotonic divergence between the two (shaded region) is a direct signature of the regime transition near $\mathcal{S}\approx 7D$ .

Figure 3: Optimization landscape of (a) displacement thickness $\delta^{*}(\mathcal{S},x_{p})$ and (b) momentum thickness $\theta(\mathcal{S},x_{p})$ across the inter-cylinder gap. Stars: LFM-derived optima; red diamonds: simulation ground truth; dashed lines indicate the downstream cylinder center ( $x_{p}=\mathcal{S}$ ) and surface ( $x_{p}=\mathcal{S}-0.5D$ ). (a) For small spacings a single minimum zone appears near $x_{p}\lesssim 1D$ ; for large spacings a second deeper minimum emerges at intermediate $x_{p}$ , marking the near-wake to co-shedding transition near $\mathcal{S}\approx 7D$ . (b) The global maximum tracks the downstream cylinder surface across all spacings, with a secondary local maximum visible near $x_{p}\approx 2D$ for higher $\mathcal{S}$ .

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

TC-VAE		Latent Diffusion
Latent dim. $d_{z}$	2	Backbone	Transformer (DiT)
Encoder	4 conv, stride 2	Attention	Factorized (spatial + temporal)
Channels	3 $\to$ 256	Temporal encoding	RoPE
$(\alpha,\beta,\gamma)$	$(0,15,1)$	Conditioning	adaLN ( $\mathcal{S}$ + $\sigma$ )
Optimizer	Adam, lr $10^{-4}$	Context frames $k$	2
Batch size	256	Framework	EDM

Appendix

Table 2: LFM architecture and training hyperparameters.

Figure 4: Time-averaged streamwise velocity $\bar{u}$ (rows 1–2) and streamwise Reynolds stress $R_{uu}$ (rows 3–4): simulation (rows 1, 3) versus LFM (rows 2, 4) for $\mathcal{S}/D=4,8,9.5$ (from test dataset). The LFM faithfully reproduces the mean wake topology, recirculation zones and wake-recovery structure, as well as the turbulent stress distribution, across the full spacing range.

Agent	Reasoning	Context inputs	Output
Planner	Direct LLM call	CSV summary, $x_{p}^{*}$ jump flags	JSON: suggested_ds , done
Analyst	Deterministic	Planner’s JSON	CSV rows (all $x_{p}$ per $\mathcal{S}$ )
Critic	ReAct loop	CSV via tool call	JSON: status , refinements
Writer	ReAct loop	CSV + figures via tools	Markdown report

Appendix

Table 3: Agent configurations. The Planner and Critic perform LLM reasoning, the Analyst executes deterministically.

AI Agents for Scientific Discovery

AI agents for scientific discovery use language models, tools, and experiments to generate, test, and refine scientific knowledge.

See all 284 papers

Large language models for partial differential equation workflows
Aug 4, 2026Han Wan, Rui Zhang, Hao SunPDE Discovery Large Language Model-Guided Optimization
Gaoling School of Artificial Intelligence, Renmin University of China, Beijing 100872, China
See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs
Aug 28, 2026Sarang Manoj Pekhale, Amartya Roy, Rajat Sarkar +1PDE Discovery Partial Differential Equations
Department of Applied Mechanics Indian Institute of Technology Delhi, India Yardi School of Artificial Intelligence (ScAI) Indian Institute of Technology Delhi, India
FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations
Oct 18, 2026Langchen Liu, Junlin Luo, Min Zhu +2PDE Discovery PDE Foundation Models
Department of Statistics and Data Science, Yale University, New Haven, CT 06511, USA · Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA · Department of Chemical and Environmental Engineering, Yale University, New Haven, CT 06511, USA

Aug 4, 2026J/K move · Enter open · S save

Large language models for partial differential equation workflows

Han Wan, Rui Zhang, Hao Sun

Gaoling School of Artificial Intelligence, Renmin University of China, Beijing 100872, China

PDE Discovery Large Language Model-Guided Optimization Large Language Model-Guided Scientific Discovery LLM Code Generation

Open paper AskPDF arXiv

Numerical methods and deep learning have advanced the modelling and simulation of systems governed by partial differential equations (PDEs), while formulating equations, configuring solvers, and translating simulations into design and control decisions continue to demand substantial domain expertise and computational resources. Large language models (LLMs) offer opportunities to automate and coordinate these tasks by combining scientific knowledge, mathematical reasoning, code generation, and tool use. Here we review LLM-assisted PDE research across three stages, including discovery, solving, and optimization. We examine how LLMs interact with physical knowledge, numerical methods, and computational feedback to formulate models, implement solution procedures, and guide design and control. We also review existing benchmarks and evaluation practices for assessing scientific validity, computational performance, and workflow reliability. Finally, we discuss three interconnected challenges and opportunities: developing mathematical and physical reasoning tailored to specific PDE problems; building transferable domain expertise through domain-specific and multimodal post-training; and integrating active experimental design with scientific validation for open-ended problems. These directions outline a path towards a more substantive role for LLMs in developing and testing scientific ideas and computational methods across PDE research.

Abstract

AI Agents for Scientific Discovery

Explore similar work

Large language models for partial differential equation workflows

See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs

FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations