PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
Authors: Mauricio Soroco
Jialin Song
Mengzhou Xia
Kye Emond
Weiran Sun
Wuyang Chen
Venue: ICML 2025 poster (2025)
Keywords: AI-for-Math
Large Language Model
Partial Differential Equation
Primary area: deep_learning->large_language_models
TL;DR: We build an LLM termed "PDE-Controller" that can achieve reasoning and planning on PDE (partial differential equation) control problems.

Abstract: We present PDE-Controller, a framework that enables large language models (LLMs) to control systems governed by partial differential equations (PDEs). Traditional LLMs have excelled in commonsense reasoning but fall short in rigorous logical reasoning. While recent AI-for-math has made strides in pure mathematics, areas of applied mathematics, particularly PDEs, remain underexplored despite their significant real-world applications. Our approach enables LLMs to transform informal natural language instructions into formal specifications, and then execute reasoning and planning steps to improve the utility of PDE control. We build a holistic solution comprising datasets (both human-written cases and 2 million synthetic samples), math-reasoning models, and novel evaluation metrics, all of which require significant effort. Our PDE-Controller significantly outperforms the latest open-source and GPT models in reasoning, autoformalization, and program synthesis, achieving up to a 62% improvement in utility gain for PDE control. By bridging the gap between language generation and PDE systems, we demonstrate the potential of LLMs in addressing complex scientific and engineering challenges. We promise to release all data, model checkpoints, and code upon acceptance.

PDE-Controller:
LLMs for Autoformalization and Reasoning of PDEs
Mauricio Soroco" "1
Jialin Song"
Mengzhou Xia
Kye Emond" 34
Weiran Sun
Wuyang Chen
Abstract
Scientists
Natural Language Instructions:
While recent Al-for-math has made strides in pure mathematics, areas of applied mathemat- ies, particularly PDEs, remain underexplored de- spite their significant real-world applications. We present PDE-Controlle ables large language models (LLMs) to control systems governed by partial differential equations (PDEs). Our approach enables I.LMs to trans- form informal natural language instructions into formal specifications, and then execute reason- ing and planning steps to improve the utility of PDE control. We build holistic solution com- prising datasets (both human-wrinde written cases and 2 million synthetic samples), math-reasoning mod- els, and novel evaluation metrics, all of which require significant effort. Our PDE-Conte Controller sig- nificantly outperforms prompting the latest open- source and GPT models in reasoning, autofor- malization, and program synthesis, achieving up to 62% improvement in utility gain for PDE control. By bridging the gap between language generation and PDE systems, we demonstrate the potential of LLMSs in addressing complex scien- tific and engincering challenges. We release all data, model checkpoints, and code https: //pde- cont relrer...rrrrrorrrr...rr..eeelll .eeer.gitttn github. io/.
control temperature the material be lower than 330."
Informal Problems
framework that en-
LLMs
Formal Specifications G(Vx:((x) < 330)
Problem Reasoning
du du dt 0x
Control PDEs (e.g. Heat Equation) for scientific problems.
Physics
Mcchanics
Figure I: We build LLMs for automated, accelerated PDE control. significant weaknesses in complex domains. LLMs strug- gle with precise mathematical reasoning (Mirzadeh et al., 2024; Feng et al., 2024; Ahn et al., 2024), understanding nuanced constraints (Williams Huckle, 2024), or making decisions grounded in ahysical-world consequences (Wang et al., 2024; Jia et al., 2024; Cheng al., 2024). Addressing these limitations will require enhancing LLMs with external tools or domain-specific reasoning.
Recent advancements in Al-for-math (Lu et al., 2022; Li ct al., 2024) have significantly enhanced LLMs in logical, formal, and quantitative reasoning, particularly for pure mathematics (geometry, probability, calculus, algebra, num- her theory, and combinatorics). These efforts address chal- lenges from grade school math (Cobbe et al., 2021) to the In- ternational Mathematical Olympiad (Hendrycks et al., 2021; Trinh et al., 2024). However, the advancement of LLMs for applied mathematics, such as partial differential equations (PDEs), remains underexplored. Unlike pure mathematies for abstract theory, applied mathematics directly addresses practical challenges, bridging theory and real-world needs. For example, PDEs are fundamental in modeling physical dynamics (aerospace engineering, quantum mechanics, fluid dynamics), providing framework to understand and con- trol systems. Integrating LLMs into applied mathematics, particularly for PDE control, holds substantial potential for advancing scientific and engineering applications.
1. Introduction
Recent advancements have significantly enhanced capa- bilities of Large Language Models (LLMs) (McKinzie et al., 2025; Huang et al., 2023). LLMSs possess pretrained common knowledge and solves daily life tasks that require commonsense reasoning without domain-ssecifii expertise. However, this reliance on generalized knowledge exposes
'Equal contribution 'School of Computing Science, Simon Fraser University 'Department of Computer Science, Princeton University DDepartment of Mathematics, Simon Fraser University "Department of Physies, Simon Fraser University. Correspondence to: Wuyang Chen <wuyang sfu.ca>.
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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PDE-Controlle:: LLMs for Autoformalization and Reasoning of PDEs
Solving PDE control problems has never been easy. Tradi- tional approaches like optimization (McNamara al., 2004) and formal methods (Alvarez, 2020) suffer from two bot- tlenecks: First, manual PDE control requires significant human efforts to understand problem descriptions and for- malize into specifications (Alvarez, 2020). Second, PDE control requires highly specialized knowledge on both cod- ing and physics, which are challenging even for seasoned mathematicians and engineers. The alternative is to lever- age pretrained LLMs. However, commonsense logic cap- tured by popular language datasets (Gao et al., 2020; Weber et al., 2024) largely deviates from scientific reasoning that requires math and physics backgrounds, leading to poor per- formance (Table 4 and 9). Moreover, unlike conventional programming languages (Python, Java, etc.) that are rich on GitHub (Chen et al., 2021), formal languages and special libraries required by PDE control (Fig. 4) are mainly used by relatively few mathematicians and engineers, resulting in limited language datasets. Thus, our core question is:
J
Temperature
Heat Equation pc au a'u ax
Heat Source ④ (time-variane)
at
au
"The temperature between x, and X2 must be within 300 30."
(1.0) g()) 8x
How to control the heat/force to satisty the constraint(s)?
Constraints
"The section between x, and X2 must be stretcheedcoopprsssedd according to kr + b."
Wave Equation 8tu 0²u P att 0x²
Displacerment
Force [time-variant]
X 0
Fax (L. = F(t)
Figure 2: PDE control adjusts inputs (heat, force) to ensure sys- tems (modeled by PDEs) satisfy spatiotemporal constraints.
2. Preliminaries
2.1. Background of PDE Control
Partial differential equations (PDEs) model nearly all of the physical systems and processes of interest to scientists and engineers. PDE control involves adjusting external inputs like heat or force to guide a system governed by physical laws (PDEs) to meet specific goals or constraints. For ex- ample, heat flow and mechanical stretchingccomppessionn in a rod are modeled and controlled by the heat and wave equations (Fig. 2). The goal is to maintain the rod's tem- perature or deformation within safe range, which requires precise, time-varying control of the heat source and applied force. This is challenging because PDEs describe complex interactions across space and time, and small changes at one point can affect the entire system. Essentially, PDE control ensures the system behaves predictably and stays within desired limits. We provide more details in Appendix A.I.
Can LLMs control PDEs with scientific reasoning?
In this work, we aim to advance Al-for-math for PDEs, making open-loop PDE control automated and accessible to broad scientific practitioners with reduced efforts (Fig. 1). Our framework, PDE-Controller, provides affirmative an- swers. PDE-Contren DDE-Controller enables autoformalization of in- formal PDE control problems into formal specifications and executable code, and enhances scientific reasoning by proposing novel subgoals to improve open-loop control util- ity. This leads to new methodology for integrating LLMs into scientific computing. Our technical contributions are summarized below:
1. Scientific Reasoning. Our EDE--Conrolller achieves up to 62% improvement in PDE control utility gain, com- pared to prompting the latest LLMs. This demonstrates promising approach to future LLM-based scientific reasoning. Our Controller is trained via reinforcement learning from human feedback (RLHF), with rewards derived from PDE simulations labeled as win or lose.
2.2. Formal Methods for PDE Control
Signal Temporal Logic. Following previous works, we use signal temporal logic (STL) (Maler & Nickovic, 2004; Alvarez, 2020) to formally represent constraints in PDE control problems:
2. Autoformalization and program synthesis. We train LLMs via supervised fine-tuning (SFT) to automatically formalize PDE control problems, transforming informal natural language descriptions into formal specifications (over 64% accuracy) and executable programs that inte- grate with external tools (over 82% accuracy).
-= T (Ve E [21, u(x) S (ar + b))
where T { and & indicates a choice from
(<,>
- Specifically:
Each STL ó defines a spatiotemporal constraint on the target variable u (the quantity to be controlled, like tem- perature, displacement). For simplicity, we only consider time-invariant linear constraints ar + b. For example: Va E 22), Vt E [ti, t2]. u(z, t) - + +00) > -3.
3. New Datasets. We build the first comprehensive datasets for PDE control designed for LLMs, including over 2 million samples of natural and formal language, code programs, as well as PDE control annotations. We also collect manually written samples by human volunteers to evaluate LLMs in real-world scenarios. Our novel dataset will serve as a high-quality testbed for future research in AT for PDE reasoning.
G ("globally")) means the constraint holds during a spec- ified interval. F ("eventually"") means the constraint is satisfied at least once during the temporal interval.
Composing multiple constraints can form more compli- cated constraints.
2

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PDE-Controlle:: LLMs for Autoformalization and Reasoning of PDEs
Init. Condition
$3.3 Autoformalization
93.2 Data
Translator
Formal Language (FI.) STL (Signal Temporal Logic)
PDE-Constrained Optimization via External Solver
Informal Problem
Reasoning Step
Natural Language (NL.
Coder
Prompt:
#3.5 Program
Instruction
Synthesis
GUROBI OPTIMIZATION
Controller
Subgoal STL Proposal
Problem
iii. End of
Control
53.4 Mutti-Step Reasoning
Utility
RLHF
Figure 3: Overview of our Con-roooerl framework. The Translator directly autoformalizes an informal PDE control problem (yellow) into formal specifications with STL (blue). The Controller proposes novel STL subgoals (purple). Each STL is synthesized into
specialized Python programs by the Coder (green) ) and optimized externally (white). From the initial condition (i.), our PDE reasoning optimizes subgoal (ii.) before the original problem, improving the utility at the end of control (iii.. We train the Controller with reinforcement learning from human feedback (RLHF).
Table 1: Overview of our LLM components. "NL.": natural language. "STL." Signal Temporal Logic.
LLM
Purpose
Training Method
Input
Output
Evaluation Metric (s)
Translator
Autoformalize constraints from informal NL into formal STL.
InU: true vs.
SFT
NL
STL.
(Sec. 3.3)
predicted constraints
Controller
Propose subgoal STLs as intermediate reasoning steps to improve the final control utility
Suceess Rate (P)
RLHE (DPO)
NL STL
Subgoal STL
(Sec. 3.4)
Utility Gain (Ar)
Coder
Generate Python code for PDE solver (Gurobi)
Exccutability (%).
SFT
NL + STL
Python code
(Sec. 3.5)
Utility RMSE r(o))
Representing PDE constraints using STL can clearly and precisely express complex specifications into logical formu- las and thus remove possible ambiguity in informal natural language. In addition to binary semantics ("satisfied" or "unsatisfied") defined above, STL admits continuous seman- al., 2009; Donzé Maler,
For more examples, please read Appendix B.
2.3. Optimization
To solve the PDE control problem defined by initial con- ditions and STL constraints (Eqn. 1), the PDE is first dis- cretized into set of difference equations with the finite element method. Then, together with STL constraints, they are formulated into a mixed integer linear program (MILP) (Sadraddini & Belta, 2015; Alvarez, 2020) where the utility r can be optimized via external optimizers like Gurobi (Gurobi Optimization, LLC, 2024). After the op- timization, if r 0, that means the system successfully satisfies & using the control input (i.e. constraints are suc- cessfully met). This is a non non-convex optimization problem. Please see Appendix A.3 for detailed formulations.
ties as utility (Kress-Gazit 2010). The utility an STL & achieves (via simulation and optimization) can be denoted as r()) E R. Please refer to Appendix A.2 for the calculation of (()).
Problem Example: Heat Equation. Consider a metallic rod of 100 mm. The temperature at one end of the rod fixed at 300K, a heat source applied to the other end. The temperature of the rod follows heat equation. We want the temperature of the rod to be within 3K of the linear constraint (xx) = I + 300 at all times between 4 and 5 seconds between 30 and 60 mm. Furthermore, the temperature should never exceed 345K at the point where the heat source is applied (x = 100). We can formulate this specification using the following composite STL formula:
3. Methods
Our core aim is automate the formalization and reasoning of PDE control problems using large language models.
Φ -G1,5) (Vxr E [30, 60] : u(x)=
( + 303)
< 0) A
3.1. Overview
Yvr E [30, 60) u((x)
( + 297] +297
> 0)) V
Problem Definition. As introduced in See. 2, the input of PDE control problem is the natural language describing the initial state, system conditions, and target constraints. The solution is a time-variant trajectory of the control input. The final output we collect is the control utility r()) E R.
G (Vx [100, 100] u(x) - 345 < 0).
3

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FoE-Contrll LLMs for Autoformalization and Reasoning of PDEs
Informal Problem
Formal Problem
Utility
Natural Language (NL)
Glaa (( [10,60)] a(x) C2 215) <o)n
Prompt:
deseriptian partir differential
natural longlungh
(fr gas od sute) =2m) >9)]
Cepeeuun (K)
Instruction eguation optimiietteo problems. Transiate the problem into Loter code rollowing spatal- signal temporat iogic."
Translator
Guaite e [100, (00 u(e) 345 < 9).
Target Problem
Formal F Language (FL):
STL (signal temporal logic) in LaTeX
Constraints
The temperature staying within 3K of p(x) =
x/4 + 300 between 4-5 a for E [30, 60] mm. The temperature must not exceed 345K at
Location (mm)
100. Consider a 100mm rod with one end at 300K and heat source at the other. The temperature follows a heat equation.
Python
meett, sart -Lentt Coddee
PDE-Constrained Optimization via External Solver
Coder
System Descriptions (Geometry, POE, Jnitial/Bouddery Conditions)
GUROBI
Net (or
Figure 4: Workflow for supervised fine-tuning (SFT) of autoformalization (Translator LLM) and program synthesis (Coder LLM). Note that the utility is only used for evaluation and not used for SFT. Without reasoning, the Translator and Coder try to faithfully and directly solve the original problem.
Framework Overview (Fig. 3). We automate the PDE
1) Constraints: Principled Syntax Formats via STL. We first design and organize eligible syntax formats for generating STLs in two levels:
control problem with the following four steps:
1. Input. Our LLM prompt is composed of 1) an instruction (to prompt the I.I.M to formalize the problem, perform reasoning, or synthesize Python programs) and 2) the tar- get PDE control problem. We build a large-scale dataset to support diverse inputs and prompts (Sec. 3.2).
For each constraint, the format in Eqn. leads to differ- ent syntax formats ((G, F} x (,,
We consider STLs up to constraints. For 2 constraints, we connect Φi and Φ22 with logical connectives or V. For 3 constraints, we consider both logical connectives and operator precedence via parentheses: d1 V d2 V Φs. Φ1 A 62 /2 (ø) V di V ((22 (1)2 (eh AΦ2) di A (øz V Φs). In total, STLs with 1, 2, 3 constraints will respectively result in 6, 72. 1296 unique syntax formats, i.c. 1374 in total (LaTeX in Fig. 4 middle top).
2. Autoformalization. Our Translator LLM will extract in-
formation about the problem constraints from the prompt, and formalize these into STLs (Sec. 3.3).
3. Reasoning. Before directly solving the target PDE con- trol problem, our Controller LLM will propose novel STLs as subgoals. The aim is to better solve the original PDE control problem by leveraging this subgoal as an intermediate step. i.e., we will first solve the problem defined by this subgoal which leads to new system state, then solve the original problem (Sec. 3.4).
At this moment, these STLs are still abstract, with hyperpa- rameters like a, b, x1, T2, t1, t2 in Eqn. 1 not yet realized.
4. Program Synthesis. Our Coder LLM will take both the prompt and the formalized STLs as inputs, and generate Python code to be fed into the external tool (Gurobi optimizer (Gurobi Optimization, LLC, 2024)) to solve the PDE control problem (Sec. 3.5).
2) System Descriptions: Sampling Initial Conditions and Hyperparameters. To realize each STL format into concrete problem, we need to fill numerical values for its initial conditions and hyperparameters (blue highlight in Fig. 4 left). We describe our sampling distributions in Appendix C.1 for the following aspects:
3.2. Principled Data Synthesis with Augmentations
Initial conditions: the initial temperature or displacement of the system at the beginning of the PDE simulation.
Training LLMs for PDE problems requires diverse data, but existing math reasoning datasets (Cobbe et al., 2021; Yang
Simulation domains: the spatial range Zmax and temporal range maxx-
et al., 2024; Glazer et al., 2024) lack large-scale corpora for PDEs. We collect key representative problems for PDE control, use them as templates, and augment them to ensure sufficient quantity and diversity.
Physical properties: such as density, specific heat capacity, thermal conductivity.
Coefficients: the linear parameters (o, b) and spatiotem- poral ranges (x), X2, 4, (2) for constraints.
Overview. As shown in Fig. 4 left, each PDE control problem consists of two components: I) constraints (red background). 2) system descriptions (blue background, including PDE, geometry, initial/bound- ary conditions). We generate dataset in three steps:
3) Dataset Synthesis with Augmentations. We syn- thesize STLs into informal natural language descriptions equipped with system descriptions. For each problem, we
4

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PDE-Controlle: LLMs for Autoformalliation and Reasoning of PDEs
Table 2: Our dataset for autoformalization and program synthesis.
given tokens in the informal input (NL). It is defined as:
Num. Constraints
1
3
Total Num.
Ctranslator SFFT
Σ log P FL;
NL; mmnssator
(2)
STLs
6
72
1296
1374
Heat (Train)
3840
45792
817776
867408
Heat (Test)
960
11448
204768
217176
Wave (Train)
3840
45504
795744
845088
3.4. PDE Reasoning via Controller LLM
Wave (Test)
960
11304
196992
209256
3.4.1. WHAT IS REASONING FOR PDE CONTROL?
also synthesize ground-truth Python code for optimization with the Gurobi solver (Fig. middle bottom). These synthesized samples give a one-to-one mapping from the informal problem (natural language) to its STL and Python code. Details about our dataset synthesis, are in Appendix C.2. More importantly, to further promote the diversity of our informal natural language problems, we use ChatGPT (GPT4o-mini) for augmentation by rephrasing without affecting semantics. In particular, we prompt five paraphrases from ChatGPT for every synthesized informal description. Please see Appendix C.3 for example of augmentation via ChatGPT.
Beyond autoformalization, an important question is: can LLMs show reasoning and planning capabilities on scien- tific problems like PDE control, where pretrained common- sense knowledge may not be helpful?
Problem Definition. As described in Sec. 2.3, the PDE control problem is non-convex. Directly optimizing the target anchor problem may lead to suboptimal solutions or intractability due to potentially poor initial conditions, loss landscape barriers, local minima, etc. Inspired by recent works in roboties (Lin et al., 2024; Wang et al.) and Al-for- math (Zawalski et al., 2022; Zhao al., 2023b;a) where subgoals are decomposed from the original problem, we propose the following PDE control reasoning strategy: The solution to PDE control problem can be improved by decomposing it into subgoals to be optimized sequentially.
As shown in Table 2, in total we have n = 2.13 million triplets of (natural language, STL, Python) samples. We merge the training set for both heat and wave problems for the training of Translator and Coder.
Real-World World Manually Written PDE Control Problems. To evaluate our LLMs on weal-world problems with high variance and noise, we collect PDE control samples de- signed by humans via questionnaires. To ensure data quality, we recruit undergraduate and graduate students as partici- pants with highly relevant backgrounds (majoring in Math, Electic Engineering. Computer Science, Physics). During four questionnaire sessions (one hour each), we provide comprehensive introductions to the settings of our PDE con- trol problems, with concrete examples and interactive com- munication. We collect 17 manually written heat problems and 17 wave problems. Details of this collection and differ- ences between our training set are shown in Appendix D.
PDE Control Reasoning. We design the following rea- soning steps for solving PDE control problems:
1. Given a target PDE control problem & (dubbed "anchor"), we sample its STL constraints into a subgoal STL, s of different spatiotemporal constraints.
2. We directly solve the anchor problem (o) and collect its utility r(ő) and runtime cost t.
3. We solve o' + Φ: We optimize d', apply the system state as the new initial condition! solve the anchor problem ø, and collect the final utility r(glo²).
4. If r()lo') > r()) (the utility of directly solving d), we call ø' a successful reasoning.
3.3. Autoformalization
5. We repeat the above steps multiple times calculate the expected performance gain.
After building our dataset, we train our Translator to extract constraints from informal natural language and autoformal- ize into STLs. Our LLM needs (1) separate constraints from system descriptions, (2) align informal definitions and concepts to formal STL syntax, and (3) connect multiple constraints with correct logic operators. This task is further complicated by context changes in different PDEs.
Whether the subgoal reasoning step can be satisfied (i.e. whether or not r(ø') > O or not) is not our concern. All we pursue is a new initial condition of the system that can better solve the original problem. As such, the time constraints of the subgoal should apply in the period prior to the anchor constraints and within the global time frame. Please see Fig. 6 for concrete examples where r(øld') outperforms r()), and more examples in Appendix B.
We leverage a pretrained 7B (Lu et al., 2024) checkpoint (MathCoder2), to fine-tune using LoRA (Hu et al., 2021) and supervised fine-tuning (SFT) with the cross-entrops loss. This measures the error in predicting each token in the output formal sequence (FL)
'To avoid long runtime of subgoal STLs, we set the Gurobi runtime threshold to 120 seconds for solving 6'.
5

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PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
3.4.2. BUILDING THE CONTROLLER LLM.
Prompt:
At
Having defined PDE control reasoning in Sec. 3.4.1, our question is: How develop LLMs perform PDE control reasoning and automatically decompose subgoals?
Instruction
Problem
u (0)
r(t)
Coder Salver
u S
At1 t'
In this section, we explain how to train a Controller LLM via reinforcement learning with human feedback (RLHF) that can perform PDE control reasoning. We illustrate our training strategy in Fig.
Controller
At2
Subgoal STL
u2'
Proposal
(i ((w)
.. ((0)
Preference Data
1) Preparing Preference Dataset. We first build a dataset of paired STLs (d/(te,, (d'()) as win-lose pairs, where e'(w) is preferred over d'(1), This dataset is used to train our Controller LLM via RLHF To build this dataset, given anchor (target) STL o, we randomly sample o' based on d, solve both, and collect their utilities.
RLHF
On 4R
((w)
Figure 5: Learning PDE control reasoning via RLHF Given the imput prompt, our Controller I.IM trained with preference data via reinforcement learning, will propose a subgoal STL s'. From the initial condition u the PDE system is controlled by d' reach state u', and then further controlled by the original STL Φ reach the final state a" We expect the utility achieved via this reasoning, r(el"") to outperform r(ő) achieved by directly solving Φ.
is up to human preference to determine which STL is favored. In this work, we prefer r (()l)())) > r(Ø)= > o0())
In total, we collect 10772 pairs of won-lose STLs.
and Gurobi optimizer to solve the PDE control problem.
2) Fine-tuning Controller LLM with RLHE During training. our Controller loads the Translator's pretrained checkpoint (inheriting the ability to faithfully formalize nat- ural language into STLs). We train Controller with Direct Preference Optimization (DPO) (Rafailov et al., 2024). Un- like the Translator's 's direct formalization into true STLs, we prompt our Controller to make modifications in propos- ing new STL variants. We leverage win-lose pairs our preference dataset as feedback to fine-tune the generative distribution of the Controller shift to the preferred STL:
Conder
Σ log P Code; FL; NL, ouder t=1
(4)
Similar to Translator, we merge the training set for both heat and wave problems for the Coder's fine-tuning.
4. Experiments
We study 1D heat and wave problems as pioneering show- cases. All our models are fine-tuned from MathCoder2, and we compare against few-shot evaluations of MathCoder2, GPT 4o, and GPT ol-mini (Achiam et al., 2023). Please read Appendix E for model and training details.
P ((O NL
CDP
- P(NL, ,010) log a Bog Prof (w)
NL)
P (0:00) NL) NL.)
)] Allog P (FL.NI.)
4.1. Accurate Antoformalization and Program Synthesis
3 log Prot (Ør()
We first evaluate the performance of our Translator for autoformalization and Coder for program synthesis.
is the sigmoid function. We load the pretrained check- point of our Translator LLM as the frozen reference model ("ref"", which serves as the KL divergence target in DPO training. P and Pref indicate the generative probability of Controller and Translator respectively (0 omitted for sim- plicity), controls the deviation of the Controller from the reference Translator. To further avoid the degradation of Controller's generation, we regularize it with the SFT loss, with weight of X (Pang et al., 2024), which we also found prevented overfitting compared to the DPO loss alone.
Evaluation Metrics. To progressively evaluate the perfor- mance of our Translator and Coder LLMs in decoupled and fine-grained manner, we propose to leverage multiple metries for different purposes, as summarized in Table 3. Among all metrics, the "Utility RMSE" (lower the better) is the most important final performance. However, it is im- portant to emphasize that we can only calculate RMSE for executable Python programs. Consequently, if we observe a coder achieving a low RMSE but also low executability. it still implies poor quality.
3.5. Program Synthesis
Forr example, switching from > < will lead to completely different constraint, but the STL itself still a valid logic.
Finally, taking both the prompt (instruction plus the target PDE control problem) and the formalized STL as inputs, we train a Coder LLM with supervised fine-tuning (SFT) to synthesize Python programs provided the PDE simulator
'Remarks for Table 4 and 5: 1) Since loU E [0, 1]. we set
the loU of any invalid STL as 0. 2) To isolate the evaluation of program synthesis without being distracted by possibly generated bad STLs, ss, when calculating the exccutability and utility RMSE,
6

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PDE-Controlle: LLMs for Autoformalization and Reasoning of PDEs
Table 3: Metrics for the evaluation of autoformalization (Transla- tor) and program synthesis (Coder). Purpose
Table 5: Autoformalization and program synthesis on manually written data (Sec. 3.2). Deviations over seeds are in parentheses. Bold indicates the best, underline denotes the runner-up. up.
Metric
LLM- -egeerrted STL (g) could be valid, but there might be se- mantic mistakes".
IoU: Intersection over union (satisfying areas) between the true STI,, &, and the LLM's generation, şi.
PDE
Model
loU (t) (Translator)
Executability (1)
Utility
(Coder)
RMSE (L)
Ours
0.7108 (0.0043)
0.8235 (0.0)
2.4687 (0.0)
MathCoder2
0.3383 (0.068)
0.9804 (0.0277)
0.0004 (0.0005)
Heat
GPT (40)
0.4314 (0.3328)
1.8555 (0.0136)
Python programs generated by Coder may not be runnahle due to bugs.
Executability: The ratio of executable programs to the total. This doesn't I ensure the executed result (utility r) is correct.
GPT (ol-mini)
0.3530 (0.3050)
0.1738 (0.0207)
Ours MathCoder2
0.6493 (0.0)
1.0 (0.0)
0.0119 (0.0)
0. (0.045)
1.0 (0.0)
0.0129 (0.0012)
Wave
GPT (4o)
0,5882 (0.4437)
0.0105 (0.0)
GPT (ol -mini)
0.3922 (0.4160)
0.0098 (0.0)
Compare the final PDE control utility r(ő) true utlity r()).
Utility RMSE: Relative
mean square error on utility.
Generalization to Unseen STL Formats. Although we only collect data with no more than three STLs, it is still possible to scale up our method to more constraints. To eval- uate our generalization to unseen STL formats, we extend our method to unseen c-constraint STLs. We evaluate our Translator (IoU) and Coder (executability & utilityy over 5 problems each for heat and wave problems in Table 6, and compare against MathCoder2. We can see that our method generalizes much better than MathCoder2. Neither model has seen problems with 4 STL constraints. MathCoder2 is evaluated with 2 in-context examples of 4 constraints.
Results The autoformalization can be evaluated using the intersection over union (loU) between the predicted and target STLs (constraints). The code generation should aim for high executability and low utility RMSE simultane- ously". As in Table 4, our Translator and Coder achieve the best across all metrics with low deviations, indicating strong and reliable autoformalization and program synthesis.
We further evaluate manually written problems. As shown in Table 5, the autoformalization may produce STLs with worse quality (lower IoU), suffering from noisy and unstruc- tured texts written by humans. Our Translator and Coder generally outperform other baselines, with the accuracy of autoformalization over 64% (STL's IoU) and program syn- thesis over 82% (code exccutability). Importantly, despite the low utility RMSE of GPT ol 01-min,, since it suffers from poor executability (only 39.22%), its RMSE cannot faith- fully characterize its stability in the real world.
Table 6: Test different models on unseen 4-constraint STL formats, based on IOU, Executability, and Utility RMSE, Bold indicates the best. Deviations over 5 seeds are in parentheses.
IoU (?)
Executability (t)
Utility
PDE
Model
(Translator)
(Coder)
RMSE (L)
Ours
0.934 (0.0)
0.8 (0.0)
0.0 (0.0)
Heat
MathCoder2
0.8154 (0.0)
0.6 (0.0)
0,2600 (0.0)
Ours
1.0 (0.0)
0.8 (0.0)
0.1515 (0.0)
Wave
MathCoder2
0.9690 (0.0)
0.8 (0.0)
0.2393 (0.2268)
Table 4: Autoformalization and program synthesis. Deviations over 3 seeds are in parentheses. Bold indicates the best, underline denotes the runner-up.
Ablation Comparisons. We further demonstrate the ne-
cessity of our Translator and Coder in Table 7, We observe the following by making comparisons:
IoU (t)
Exccutability (t)
Utility
PDE
Model
(Translator)
(Coder)
RMSE (L)
Ours
0.992 (0.07)
0.9978 (0.0015) 0.0173 (0.0065)
MathCoder2
0.772 (0.35)
0.9592 (0.0166)
0.2058 (0.0672)
Our Translator has better autoformalization abilities, im- proving +28.5% IoU over MathCoder2.
Heal
GPT (4o)
0,5807 (0.3244)
0.0445 (0.0437)
GPT (o1-mini)
0.3561 (0,2857)
0.0898 (0.0165)
Ours
0.992 (0.07)
0.9620 (0.0098)
0.0076 (0.001
Our Coder is robust to noisy autoformalization:
MathCoder2
0.772 (0.35)
0.9340 (0.0242)
0.1089 (0.0485)
Wave
GPT (4o)
0.6799 (0.2523)
0.0868 (0.0500)
Given ground truth STLs, our Coder has +4% better ex- ecutability rate in generated Python than MathCoder2, and utility RMSE is 91.6% lower (better) than Math- Coder2.
GPT (ol-mini)
0.4041 (0.2771) 0.0757 (0.0149)
LLMs are provided with true STLs in their prompts, instead of LLM- generated STLs. Table 15 and 16 show end-to-end results but cannot decouple the quality of program synthesis. 4) The utility RMSE is only calculated for executable Python programs. 'Otherwise, the code generation might trivially generate for example runnable "return 0", which is obviously incorrect; it might get lucky and generate the correct code for only a very limited set of problems.
When switching from ground truth STL inputs to noisy Translator predictions, our Coder's utility RMSE is only 0.57% worse, indicating that our Coder is robust under noisy predicted STL inputs.
7

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PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
Initial Condition
Anchor r(1) - 0.81
Reasoning
Heat
t'= = 1.49
9.86, 3.48
Temperature (K)
420.
(0)
u
u²
350.
6
Φ +
300.
A
250
40.0
60.0
100.0;
20.0
60.0 80.0 100.0
40.0
60.0 80.0 100.00.0
20.0 40.0
100.0
Location (mm)
Location (mm)
Location (mm)
Location (mmm)
Wave
t
0
0.92, (()) 2.02
0.11
0.92, r())d') r 3.55
Disp Displacement (mm)
0' +
0
0.0
20.0
40.0
60.0
80.0
100.0
0.0
20.0
40.0
60.0
80.0
100.0
0.0
20.0
40.0
60.0 0 80.0
100.0
0.0
20.0
40.0
60.0 80.0
100.0
Location (m)
Location (m)
Location (m)
Location (m)
Figure 6: Case study of LLM reasoning for PDE control heat (top) and wave (bottom) problems (symbols aligned with Fig. 5), From left to right: Directly solving & from the initial condition (") (Ist column) yields r()) (2nd column); Reasoning: solving o' from (0)) to get u' (3rd columa) then solving é from u' to get r(ole') (4th column). Black curves indicate the system's states (temperature for heat, displacement for wave) at L or z'. Colored segments (A, B, C) are constraints. with dashes for inequalities (2 when dashes are above the solid, and vice versa), Although we plot constraints, they may constrain temporal ranges (1i, f2] $ t', Constraint STLs can be found in Appendix B.3.
Table 7: Ablation comparisons of translators and coders. Bold indicates the performance change relative the row above.
Table 8: Overview of our reasoning data. We threshold 3 difficulty levels of questions by the Success Rate P of random sampling.
Heat
Training
Testing
Total
Method
Metric
Performance
Mathcoder's 's Translator (Table 4)
0.772
Num. (1)()) Pairs
4813
1181
5994
Translator (Table 4)
(t)
0.992 (+28.5%%)
Easy P E (0.8, 1)
27.1%
26.1%
26.9%
Mathooder's 's Coder (Table 4)
0.9592
Medium P E (0.5, 0.8]
37.3%
37.8%
37.4%
Our Coder (Table 4)
Executability (1)
0.9978 (+4.02%)
Hard Pe E 0.,0.5
35.6%
36.2%
35.7%
Mathcoder's Coder (Table 4)
0.2058
Wave
Training
Testing
Total
Ulility RMSB (1)
Our Coder (Table 4)
0.0173 (-91.6%)
Num. (d(()),d(()) Pairs
3812
966
4778
Translator STL + Coder (Table 15)
Ground Truth STI. Coder(Table Utility RMSE (L)
0.0174
Easy P E (0.88, 1)
32.5%
33.6%
32.7%
0.0173 (-0.57%)
Medium P P (0.55, 0.88)
33.1%
32.5%
33.0%
Hard P E (0, 0.55
34.4%
33.9%
34.3%
4.2. Improved Utility via PDE Reasoning of Controller
problems. A (r(ll'')) - r(d)]-
Beyond autoformalization and program synthesis, our most important contribution is the scientific reasoning on PDE problems by the Controller. In addition to the MathCoder2 and GPT models, we consider another baseline, random sampling, which naively generates reasoning steps by ran- domly sampling the anchor's constraints.
Difficulty Levels. Intuitively, some anchor problems easy to improve via reasoning, while others may be more challenging to improve. To comprehensively study the per- formance of our controller, we design three difficulty levels based on the Success Rate, P, of random sampling. We group problems by choosing thresholds on of random sampling such that all difficulty levels share a balanced num- ber of problems during testing. We overview our reasoning data in Table 8 and provide examples in Appendix B.
Evaluation Metrics. During inference, we sample from the Controller multiple times. We evaluate the reasoning performance on PDE control problems with two metrics:
Success Rate P: The percentage of sampled reasoning step (Φ') that can improve the anchor problem (6), av- eraged across all anchor problems. P A P(p) = Esp (r(old') > r())).
*In Table 9, to isolate the evaluation of the PDE reasoning qual- ity (ő') from influence by autoformalization and code generation. we provide the true Python code for any valid generated subgoal STL ø' Table 17 shows end-to-end results but cannot decouple the quality of L.M PDE reasoning.
Utility Gain Ar: The expected improvement in utility
via a sampled reasoning step, averaged across all anchor
8

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PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
Table 9: Scientific reasoning over PDE control problems via our Controller LLM. Deviations over seeds are in parentheses. "Valid STL e' (%)" ratio of valid proposed subgoal STL o' (i.e. without any syntax errors or improper time constraints). Bold indicates the best, underline denotes the runner-up. "x" indicates no valid STLs were generated for evaluation. indicates not applicable.
Success Rute 7 (t)
Unlity Gain Br (t)
Difficulty Level
PDE
Random
Math-
GPT
GPT (4o)
Random
Math-
GPT
GPT (4o)
Ours
Sampling
Coderv2
(ol-mini)
Ours
Sampling
Coderv2
(ol fol-mini)
Fasy
0.966 (0,0453) 0.886 (0.0365) 0.396 (0.2419)
0.718 (0. 1012)
2.233 (0.6662)
1.594 (1 (333))
0.795 (U7985)
1.614 (0.0262)
Medium
0.877 (0. 1010) D.694 (0.09199 0.340 (0.2324)
(()
0.468 (0.0912))
1,090 (0.5030)
0.526 (0.6986)
-0. (0.2659)
.0,001 (0)
0.222 (0.0560)
Heat
Hard
0.592 (0.1692) 0,356 (0.1319) 0,236 (0,1887)
(
0.469 (0.1147)
1,33 (0.84166
-0.380 (1.2688)
-0,961 (0.5101)
-1,489 (0)
0.855 (0.0611)
All
0.812 (0. (052) 0.645 (0.0864) 0.324 (0.221)
0.552 (024))))
1.453 (0.6726)
0.580 (1.0002)
-0,093 (0.5249)
0.897 (0.0477)
Easy
0.936 (6.0600) 0.928 (0 (0.0252) 0.954 (0.0725)
1 (0)
1 1.423 (0.4135)
1.110 (0.486)
1.601 (0.1794)
1,706 (1)
Medium
0.833 (0.1009)
(0.0894) 0.769 (0.0362)
0.933 (0.0408)
0.901 (0.1389)
0.704 (0.7072)
0.830 (0.2122)
0.652 (0.0191)
Wave
Hard
0.328 (0.1220) 0.294 (0. (0.16200 0.324 (0.1036)
0.386 (0. 1013)
-0.349 (0.4357)
.0.531 (0,8028)
40.670 (0. 1945)
-0.609 (0.0196)
All
0. 8.699 (0.0943) 0.653 (0.0915)
(0.0708)
0.773 (0.0474)
0.658 (0.4293)
0.427 (0.6695)
0.587 (0. 1954)
0.583 (8.0129)
Valid STL of (1)
82.70 82.70
42.45 (10.54) 0.04 (0.10)
2.55 (0.65)
82.70 (1.97)
42.45 (10.54) 0.04 (0.10)
2.55 (0.65)
Results We observe the following from Table 9:
informal math into formal logic. LLMs like GPT-3.5/4o have been used to translate problems into Isabelle/Leand sketches (Zhou et al., 2024; Jiang et al., 2023). Verna- Copter (van de Laar al., 2024) autoformalizes natural languages into STL with correctness checks. Hybrid meth- ods combine manual and automated steps (Xin et al., 2024; Xiong et al., 2023; Murphy et al., 2024; Mishra et al., 2022). We introduce LLMs trained to autoformalize informal PDE control into STL. Second, our paper targets PDE Controls. PDEs model physical systems; control aims to ensure stabil- ity or optimize behavior (Alvarez, 2020; Wei et al.). Clas- sical methods use adjoints (Lions, 1971). while learning- based ones use differentiable physies (Holl et al., 2020) or RL (Rabault et al., 2019). LLMs help with multimodal PDE surrogate modeling (Lorsung & Farimani. 2025). We reformulate PDE control into MILP via FEM (Sadraddini & Belta, 2015) and use LLMs to propose initializations for open-loop control, guided by a utility score (Appendix A.2). Finally, our work also studies I.LM-based Task Plan- ning. LLMs translate natural language to formal plans (e.g.. (Pa) et al., 2023). Recent work improves plan reliabil- ity (Wang et al.; Ren et al., 2023), with CLMASP refining outputs via ASP (Lin et al., 2024). We extend LLM--based reasoning to PDE control for the first time.
In general, our Controller consistently outperforms other models for both heat and wave problems. While second to GPT-4o by wave success rates, GPT-4o suggests very few valid STLs caused by syntax errors or time constraints that occur after the anchor problem.
In general, our Controller most significantly improves the utility. For example, on heat problems, our Ar = 1.453, which improves 62% over the second-best (GPT 4o, Ar- 0.897). Although model improves utility for "hard""-leee wave problems, our Controller still proposes the highest quality subgoals.
Our Controller suggests far and away the highest ratio of valid STLs (ø'), almost doubling the second-ranked MathCoder2.
In most heat cases, MathCoder2 and GPT models are worse than random sampling. GPT ol-mini fails entirely due to invalid hallucinations in subgoal STL proposals. On manual data, due to invalid subgoal STL proposals, all models fail to generate meaningful reasoning steps. Firstly, inconsistencies in the natural language of the manual data cause the models to generate invalid constraint values, such as different units to describe time within sentence. Sec- ondly, new notation such as ";=" in NL, result in unbalanced brackets, hallucinated numbers, and quantifiers. Thirdly, the models propose invalid time constraints for reasoning steps, which should occur before the anchor's 's time constraints.
Please read Appendix G for more related works.
6. Conclusion
In this paper, we aim to transform the automation of PDE control problems. By utilizing LLMs to interpret natural language problem descriptions, formalize mathematical ex- pressions, and apply scientific reasoning, we demonstrate that LLMS can fully automate PDE control while even im- proving control performance. Our research emphasizes the importance of L.LSs in applied mathematics, and seeks to enable more accessible, scalable, and robust PDE solutions, ultimately expanding the practical reach and reliability of PDE applications across scientific and engincering domains.
Case Study. To better illustrate our PDE controller, we show one case for heat and wave in Fig. 6. In both exam- ples, first optimizing the reasoning steps leads to new initial conditions that better solve each anchor problem.
5. Related Works
First, our paper belongs to the broader community of auto- formalization in Aa-hor-math, Autoformalization converts

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PDE-Controlle: LLMs for Autoformalization and Reasoning of PDES
Acknowledgements
Feng, G., Yang, K., Gu, Y., Aí, X., Luo, S., Sun, J., He, D., Li, Z., and Wang, L. How numerical precision af- fects mathematical reasoning capabilities of LLMs. arXiv preprint
We thank Dr. Danqi Chen for her helpful comments. We thank all participants in our questionnaire for manually writ- ing PDE control problems. For privacy reasons, we do not disclose their names.
2024.
Gao, L., Biderman, S., Black, S., Golding, L., Hoppe, T. Foster, C.,, Phang, J., He, H., Thite, A., Nabeshima, N., et al. The pile: An 800gb dataset of diverse text for language modeling. arXiv preprint arXiv: arXiv:2101.00022 2020.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none of which feel must be specifically highlighted here.
Garnier, P., Viquerat, J., Rabault, J., Larcher, A., Kuhnle, A.. and Hachem, E. A review deep reinforcement learning for fluid mechanics. Computers & Fluids, 225:104975 2021.
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
A. More Background on Formal Methods for PDE Control
A.1. PDEs
We consider controlling systems governed by two popular PDEs (in ID space):
Heat Equation: Describes how heat diffuses through a material over time. Applications: temperature in buildings, pollution in the environment.
du pc:
= 0,
Öt dx.
du
dr (L, t) = q(t),
(5)
u (0, t) 9o.
Vt E [0, max)
(xr 0) uz(
Vr E [0, L].
P,C.K. K. > O: density, specific thermal capacity, thermal conductivity of the material respectively. u the spatiotemporal
temperature of the material. x E [0, L] is the spatial location. t [0, fmax] is the time. is the time-variant external heat source, applied at L. Yo the boundary condition applied 0. un the initial condition (temperature).
Wave Equation: Models the propagation of waves (sound, electromagnetic, or water waves). Applications: Acoustic control, vibration control in structures (e.g., bridges, buildings).
Qu
duu
E
0,
012
Or.
E du
(I, t) F(t),
(6)
dr
u(0,t) t) 90.
Vt E [0, tmx)]
uxr, 0) uz(
Vr E [0, L.-
P, E' > O: density, Young's Modulus of the material respectively. u the spatiotemporal displacement of the material. x E [0, L] is the spatial location. the time. F is the varianariant external fource, applied at x L. 9o is the boundary O. uo is the initial condition (displacement), and typically we set as
condition applied
A.2. Utility of STL
The continuous utility value of STL r()) is calculated with the following cases and rules:
r (u, u ax + b,t) u(x, 6) - (ar + b)
r ru, u az + b,t) (ax b) u(x,t)
(8)
r (u, Φ2,t) min {r (u, 01, ,), (ue, Φ2, t)) r (u, Φ1 V (2,t) max (r (1, 1), (u, 02, 1)} u, (1,)) = sup
(9)
(10)
(r(u,
(11)
t)((t+,,t+))
inf r (u, (o, (E(tta.14)
(u,
(12)
Here, Eq. 7 and 8 indicates the linear constraint we consider in Eq.
A.3. MILP Formulation of Control Synthesis
Solving the PDE control problem can be relaxed and formulated into PDE-constrained optimization problem, which can be further solved by mixed-integer linear programming (MILP). We brief the high-level steps, and we recommend readers to read (Sadraddini & Belta, 2015; Alvarez, 2020) for more details.
We start with spatially and temporally discretizing the PDE. This is achieved by the finite element method (FEM):
1. The PDE is converted into its weak (variational) form by integrating against suitable test functions u(x). The purpose is to reduce the second-order derivatives of u to first derivatives so that we can obtain (linear) approximations to u. This also simplifies boundary condition handling and smoothness requirements in the original PDE.
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDES
2. The spatial domain of the PDE's weak form is discretized by dividing it into small, simple geometric elements (intervals in ID, trianglesquaadrilatrrass in 2D; tetrahedra in 3D), essentially forming mesh, and choosing local basis functions each element.
3. Contributions from local (spatial) elements assembled across the entire mesh into global linear system, often written as Mû + Kü F. The stiffness matrix (K) and mass matrix (M) encode the PDE's structure. The banded stiffness matrix (K) arises from terms involving derivatives (e.g., Vu - Ve in the weak form). The diagonal mass matrix (M) comes from non-derivative terms (e.g., u v). F is the "force" vector (from external source terms/boundary conditions like heat and force). This step transforms the PDE into an ordinary differential equation (ODE) in time for the discretized spatial domain.
The temporal domain is further discretized using finite difference schemes, obtaining a of difference equations that must be solved at each time step. This final step produces the final linear (or nonlinear) system of equations that solved numerically to approximate the solution of the original PDE.
u
M
Ku F
(13)
At
This can be further rearranged a linear system:
(M + AIK) k+1 Mu
AIFR+1
(14)
At this moment, we can re-formulate the original PDE control problem into the following PDE-constrained optimization problem:
max r (6, ú)
(15)
s.t. (M + A:K u++1== = Mu A1F
(16)
ū Ú0)
This formulation is equivalent to an MILP problem because:
Φ is only applied limited spatiotemporal ranges. After the discretization of PDE, essentially ø is only selectively applied to certain areas of grids over our (1D) mesh. That means, we need binary variables to encode the absence/presence of d over the spatiotemporal domain.
The discretized PDE constraints Eq. 16 is linear in u. Additionally, based on Appendix A.2, the objective function r(o, ú) (Eq. 15) is also linear in ø and ü.
This MILP problem non-convex, due to the min, max operation and non-differentiabiliyy (Appendix A.2) of the objective function (ø, ū) in Eq. 15. This MILP problem solved using the aff-the-shelf Gurobi solver (Gurobi Optimization, LLC, 2024).
A.4. Metries Additional Design Considerations
The careful design of our utility score, in Appendix A.2, can faithfully quantify whether the STLs (constraints) are fully met by the solution simulated by the Gurobi solver. This utility score is inherited from prior work (Alvarez, 2020).
Compared with standard metrics such as @ass@k, our metrics can better quantify our LLMs performance. IoU is designed to provide fine- mne-grained quantification of the logic structures (junctions, temporal operators, boundary constraints) while pass@k k lacking nuance will fail or succeed from trivial text-level differences STI. We further explain the connection between our metrics and @ below:
For autoformalization (Translator), IoU is equivalent to Faverage@k"" k", it averages the alignment between predictions and targets over multiple generations and problems. We can discretize loU into "pass@k" by considering only pass/fail cases based on token-level differences; but this ignores fine-grained quantifications of autoformalization. Namely, in tables 10 and 11 below, loU and pass @k are not always aligned.
For code generation (Coder), the executability metric is essentially pass@k from the executability perspective.
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDES
Table 10: Performance comparison for autoformalization (Translator) on Heat. Deviations over seeds are in parentheses.
Model
IoU
Pass@1
Pass@2
Pass
Ours 0.992 (0.07) 0.978 (0.
0.980 (0.134) 0.982 (0.131)
MathCoder2 0.772 (0.35) 0.538 (0.480) 0.565 (0.484) 0.583 (0.493)
Table 11: Performance comparison for autoformalization (Translator) on Wave. Deviations over seeds are in parentheses.
Model
LoU
Pass
Pass
Pass
Ours 0.992 (0.07) 0.971 (0.161)
0.975 (0.152)
0.977 (0.149)
MathCoder2
0.772 (0.35) 0.3305 (0.4396)
0.3726 (0.4663)
0.3971 (0.4893)
Heat
Initial Condition
Anchor
Reasoning
Easy
t
0
t 11.77, x()) = 18.39
0.99
11.77, 19.55
400.
Temperature (3)
u
4
350.
Φ +
300.
Ø'
0.0
20.0
40.0
60.0 80.0
100.0
0.0
20.0
40.0
60.0
80.0
100.0
0.0
20.0
40.0
60.0
80.0
100.0
0.0
20.0
40.0
60.0
80.0
100.0
Med.
t
0
13.41, (Ø)
33.43
2.03
13.41,
2.89
u (0)
u.'
Temperature
350.
Ø'
300
0.0
20.0
40.0
60.0
80.0
100.0:
0.0
20.0
40.0
60.0
80.0
100.0.0.0
0.0
20.0
40.0
60.0
80.0
100.0
0.0
20.0
40.0
60.0
80.0
100.0
Hard
0
t
4.50,
r())
9.10
2.50
t
4.50,r(
13.77
ee g 400
380
340
Ø'
370.
280,
0.0
20.0
40.0
60.0
80.0
100.0,
20.0 40.0 60.0
80.0
100.0;
0.0
20.0 40.0 60.0
80.0 100.00.0
20.0 40.0 60.0 80.0 100.0
(mm)
Location (mm)
Location (mm)
Location (mm)
Figure 7: Case study of heat problems with different difficulty levels: casy (top), medium (middle), hard (bottom). Symbols are aligned with Fig. 5. From left to right: Directly solving from initial condition (0) (Ist column) yields r()) (2nd column). Reasoning: solving ø' from ( to get u' (3rd column) then solving á from u' get r(øld") (4th column) Black curves indicate the system's ''s states dashes for inequalities (2 when dashes are
(temperature for heat, displacement for wave) at t'. Colored segments are constraints, above the solid, < when dashes are below the solid). Note that although we always plot constraints (segments), they actually constrain different temporal ranges (f1, to] and it is possible that f, t' & t2)-
B. Case Examples and Visualizations
To better illustrate our PDE reasoning, we show more cases with visualizations and their corresponding STLs (constraints). All time constraints are rounded to two decimal places, and parameters that describe the linear profiles are rounded to four decimal places.
B.1. Heat
We show easy/medium/hard problems Fig. 7, with their anchor STI. (0) and subgoal STL (6') listed below.
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
Easy:
Anchor Constraints STL (Φ):
F1.173344 (Va [10, 250, (u(x)) (000)))((-000000000 (-0.2169-x 286.1171) 0))A
19 (Var E [41, 55) (u(x)) (-0.2225 + 311. < 0))V
F18.04, (Vr E [61, (u(x) - (0.0988 x + 310.7904) > 0))
Subgoal STL Proposal (ø') by Controller:
920 (Va [50, (u(x) (-0.2907 323.3970) > 0))A
(G10.00..77 (Vx E [31, 46] (u(x)- (-0.1338 r 368.9958) < 0))V
F10.000.000 (Vr [61, (u(()) (0.0788 x 390.7948) > 0)))
2) Medium:
Anchor Constraints STL (Φ):
(1z E [0, (0,30((u(a)) (0.4159 x + 293.2549) > 0))V
(G14.03,7.79) (Vx E [46, 63] (u(x) (-0.0956 x 296.0596) < 0))4
13 (Vr E [75, 96] (u(x) (0.2602 309.7111) 0))
Subgoal STL Proposal (d') by Controller:
G.0. (Vz E [0, 30 (u(x)) (0.3616 x + 387.4454) > 0))V
(G10.52..00 (Vx E [46, 63] (u(x)) (-0.1061 + 435.4267) < 0))4
F10.2211111 (Va [75, (u(x)) (0.1779 x + 374.4556) > 0)))
3) Hard:
Anchor Constraints STL (Φ):
G2.62,4.00) (Vz E (22(8))(u(z) (-0.0122 x 294.2976) > 0))
Subgoal STL Proposal (ő') by Controller:
G 211.29,2.50 (Vz E (22,8))(u(z)) - --000117 - 408.1535) > 0))
B.2. Wave
We show easy/medium/hard problems in Fig. with their anchor STL (0) and subgoal STL (Ø') listed below.
1) Easy:
Anchor Constraints STL (p):
(G (G.0.25.0.54 (Væ E [7207, (u(x)) (2.2684e 05 + 1.4129) 0))A
F10.76.0.84 (Vz E [42469, 65095) (u(z) (1.8952e 06 . 1.7928) > 0)))V
F11.2.133 (Ya E [77653, 85444 (m(x) - -(-4.0675 - 05 + 2.1560) > 0))
Subgoal STL Proposal (ø') by Controller:
(G (Vo [7207, (3479[(u(x)) (3.0242e 05 1.1961) 0))A
Fo0.05.0.099 (Vx [42469, 65095 65095](u(z) (2.3575e
06 r 1.5473) 0)))
F1..0.00.8 18 (Vz
[77653, 85444 (u(z) (-4.4208c - x > 0))
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDES
Wave
Initial Condition
Anchor
Reasoning
Easy
1 1.33, r(6) 4.52
0.24
1.33, r(bld) = 7.09
Displacement (mmm)
2 (0)
u'
0.
-2
-4
0.0
20.0
40.0
60.0
80.0
100.0
1.00
20.0
40.0
60.0
80.0
100.0:
0.0
20.0
40.0
60.0
80.0
100.0
0.0
20.0
40.0
60.0
80.0 100.0
Med.
8
0
t 1.31,
r(0)
2.58
t'
0.16
8
1.31,
=(d))')
5.09
Displacement (mim)
12'
2.
0.
-2
4.
-6
A
A
80.0 100.0
0.0
60.0
100.0
0.0
20.0
40.0
60.0
80.0
100.0
0.0
20.0
40.0
60.0
80.0
100.0
Hard
t
t = 1.11, (())
3.54
0.22
1.11,
r((ld')))))))))))
5.54
Displacement (mm)
(0)
2.
-2.
-.
0.0
20.0
40.0
60.0
80.0
100.0
:.00
20.0
40.0
60.0
80.0
100.0:
0.0
20.0
40.0
60.0
80.0
100.0
0.0
20.0
40.0
60.0
80.0
100.0
Location
Location (m)
Location
Location
Figure 8: Case study of wave problems with different difficulty levels: easy (top), medium (middle), hard (hottom). Symbols aligned with Fig. 5. From left to right: Directly solving from the initial condition (9) (Ist column) yields r()) (2nd column). Reasoning: solving o' from ( to get (3rd column) then solving & from a' to get r()ld') (4th column) Black curves indicate the system's states (temperature for heat, displacement for wave) at t or t'. Colored segments are constraints, with dashes for inequalities (2 when dashes are above the solid, when dashes are below the solid). Note that although always plot constraints (segments), they actually constrain different temporal ranges [f1, ta] and it is possible that t [2).
2) Medium:
Anchor Constraints STL (0):
(Vr E [13787, 21080 (u(x) - (-9.5400e - x 0.3744) 0))V
00.05.0.00 (Vr [49923, 59039] (u(x) (1.2003e 05 1.5231) 0))V
L31.8,,13 E [78762, (u(x)) (4.3983e 05 1.5994) > 0))
Subgoal STL Proposal (ő') by Controller:
10.06.0.161 (vz e [9084, 26246) (u(x) - (-4.3491c 05 2.1348 0))A
G0o.0,00000 (Vx E 27204,,39
39168) 391688(((())) (u(x) (4.2688e 06 2.6843) > 0))A
G0..01,0077 (Vz E [58194, (u(r) (-4.0965e 05 35-r x + 0.2641) < <))
3) Hard:
Anchor Constraints STL (Φ):
(G (Vo E [12400, 20684](u(x)) (4.0369e 05 0.9002) 0))A
82..22u032 (Va e [33059, 46052 (u(x) (3.5491e
07 1.4933) < 0))]
FiL.o.1.1 (Va E [67963, (932) - (1.6090e 06 x 1.1675) > 0))
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
Subgoal STL Proposal (ø') by Controller:
(G (Vz E [12400, 20684[ (3.4256e - 05 0.5172) > 0))
F10.01.0.11 E [33059, ( (6602 - (2.9472e 07 x 1.8896) < 0)))
200.01.0.20 (Va [67963, 79313(uu(x) - (2.3060e 06 x 0.7148) > 0))
B.3. STLs for Examples in Fig. 6
Heat:
Anchor Constraints STL (d) shown as A, B, C respectivvly in the Initial Condition, Anchor and right Reasoning columns of Fig. 6:
G 111.3,3.13 (Vr E [2, 29 (u(x) (0.4565 x + 287.7909) > 0))V
(G11.25,4.599 (Var [38, (u(x)) (0.2137 287. 287.1038) 0))A
F15.94.9.86] (Vr [54, (u(x) (0.3008 + 299.3877) 0)))
Subgoal STL Proposal (ő') by Controller shown as A, B, respectively in the left Reasoning column of Fig. 6:
100 (Vr [2, 29] (w()) (0.4088 x + 294.5123) 0))V
(G10.20.0.33 (Vx [38, (u(x)) (0.2907 r 404.7615) < 0))A
10 (Vr E [54, (u(x) - (0.3503 r 316.13354 > 0)))
Wave:
Anchor Constraints STL (ő) shown A, B, C respectively in the Initial Condition, Anchor and right Reasoning columns of Fig. 6:
(G.0.16.0.200 (Vr E [14057, 29980] (u(x) (2.8994c - 05 x 2.5372) > 0))A
G10.28.0.77 (Ve [38096, 58208] (u(x) (-2.9597e 05 r 0.8070) > 0)))
F10.45.0.92 (Vz E [71793, 88339 (u(z) (1.2523e 05 2.4337) < 0))
Subgoal STL Proposal (ø') by Controller shown as A, B, C respectively in the left Reasoning column of Fig. 6:
(G0.00.0.00 [14057, 29980 (u(x) (3.0385e
1.3785) 0))A
G1o.03.0.77 (Vz [38096, 58208] (u(x) (-2.2655e
05 0.5356) 0)))
110.00.0.11 (Vz (Vz E [71793, (u3))) (1.6430e 05 1.7368) < 0))
B.4. Solutions Examples in Fig. 6
Fig.9 presents the optimized control inputs for the heat and wave problems shown in Fig.6. The figure illustrates how the heat source (g:) and the force (Fi) applied the end of the rod change over time. We provide the solutions for cases, the first column shows the solution obtained by directly solving the problem while the second column shows the solution solve the problem o after the subgoal d'
C. Dataset Details
C.1. PDE Parameter Ranges
For generating our dataset, we sample hyperparameters (that define control problems) in the following range (Table 12 and 13). Note that our heat and wave problems are all ID, making some dimension- units different from those in 3D.
C.2. Rules for Data Generation
From STL to Natural Language. Each natural language problem consists of two parts: one part defines the premises, such as the material density, initial temperature, and rod length; the other part describes the constraints, which can be expressed as an STL formula.
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
Φ Φ
1e6
1e6
1.00
1.00
(t(mW))
0.75
0.75
0.50
0.50
0.25
0.25
0.00
0.00
8
10
8
10
1e3
5.0
5.0
F (N)
0.0
0.0
-2.
22.5
55.0
-5.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
t (S)
(S)
Figure Synthesized control inputs heat (top) and the wave (hottom) problems Fig.6. Left: Solution for directly solving ø Right: Solution for solving Φ based on subgoal ø', where red vertical dashes indicate the control shift from o' to Φ.
Table 12: Ranges for hyperparameters used in heat problems.
Rod Length
Fixed Temp. (K)
Max Time (8)
Thermal Conductivity
Density
Specific Heat Capacity
Linear Profile Param.
(mm)
(mWW mon/ K)
(kyf man)
(pJ/kg/ K)
L
temp
b
(x 10%)
(x 10
(x 1x10
(x10 ")
(x 10")
(x 10")
[50, 300]
[250, 350]
[S, 15]
[-0.5, 0.5]
tevpp [-20, 20]
[1.2, 1.8
[0.4, 1.2]
[3.6]
(3.6]
[3, 4.51
[4.5, 4.8)
Table 13: Ranges for hyperparameters used in wave problems.
Rod Length (mm)
Density
Young's ' Modulus
Max Time (s)
Linear Profile Param.
(kg/ mmm)
(N)
Pated (x 10 ")
Pleass (x10
Exed (x10")
E (x10*)
Ox100
(60000, 140000]
[7.6, 8]
18.4, 8.8)
[2, 2.4]
[1, 1.8)
[0.5, 2]
[-5, 5]
[-3, 3]
To convert the STL formula into informal language, each constraint and condition is mapped to corresponding phrases. For instance, the constraint conditions F and G are described "for one point during" and "for all time between". respectively. The comparison conditions mapped based the problem type: for heat problems, > indicates "the temperature distribution of the rod should be greater than"; for wave problems, indicates "the displacement of the rod should be stretched over".
then consider the V and A logical operators after converting each individual constraint. For problems with two constraints, we introduce transition words such as "moreover" for and "either... or... for V. For problems with three constraints, we design templates that account for the hierarchy of constraints based on the placement of parentheses in the STL formula that defines the logical order.
For example, given the STL syntax (A A B) V C, the template is: "Either satisfy the conditions that A and also B; satisfy the condition that C." For A A (B V C), the template is: "Satisfy A. Afterwards, either consider B or C."
From STL to Python. To parse the predicted STL into Python in Table 9, we first extract logical connectives (V, A), rod intersections, and constraint equations. Based on the number of constraints, the corresponding variables are inserted into a Python script template. The time intervals and constraint conditions (G, F) are passed directly into the script to preserve the original STL syntax. The generated outputs may fail to convert due to hallucinations or syntax errors.
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PDE-Controlle:: LLMs for Autoformalization and Reasoning of PDEs
For example, given the STL Logic below:
G |0.049.0.053| (Vr e [9829, (99)) (1.882e 05 0.1877) < 0))A
56082 F (0.051,0.149] (Vz E [40199, 56082] (u(x) - (3.356e - + -0.500)) < 0)))V
F 00.061.0.169] (Vor E [75646, (8769) 98769((u())) - ---1.390e x +
We convert into its Python snippet:
ape0 = logic. (CCon(([9829) 19907], "<",
lambda x: 1.882e-05 xx + 0.187, lambda x: 1.882e -05)
apcl
logic.APCont APCont([401)) 56082]. "<"
-05110, lambda e apc2 logic APCont(175646, 98769], ">" lambda x: -1.390e-05 * x 2.844, lambda x: -1.390 -05)
lambda x: 3.356e-06
cregions ("A": apc0, "B": apel, "C": "p" apc2]
espec = "(((G_10.04) 0.053] (A)) &
[F_[0.051] 0.149] (B)))
(F_[0.061. 0.169] (C)))"
C.3. ChatGPT Augmentation
We adopt the data augmentation pipeline proposed by (Pan et al., 2023), where ChatGPT 4o-mini is prompted to rephrase each natural language problem five times. The problems are formulated based on STL logic with PDE parameter variables enclosed in squared brackets, which are further replaced with randomly values sampled from ranges in Appendix C.I. For example:
Prompt:
Rephrase the source sentence in 5 different ways. Make the outputs as diverse as possible. Make sure not change the equations starting with mu and parameters in the squared brackets, and maintain all the parameters while rephrasing.
Source (original problem): For one point during the time interval [t0] and [t]], the temperature distribution of the rod should be larger than the linear profile mu0(x) [a0] * + [bO] between section [rO] and [r1].
GPT Outputs:
At specific moment within the time period [t0] to [t]], the temperature distribution along the rod must exceed the linear profile mu0(x) [a0] [bo] in the segment defined by [rO] and [rl].
2. Within the time frame of [t0] to [t1], there should be at least one instance where the rod's temperature distribution surpasses the linear profile mu0(x) [a0] xx+ + [bO] from [rO] to [rl].
3. During the interval between [10] and [t]]. there is a moment at which the temperature across the rod must be greater than the linear profile mu0(x) = [a0] x + [bO] within the range of [rO] and [r1].
Throughout the duration from [t0] to [t1], there exists a point in time where the temperature distribution of the rod should be greater than the linear profile mu0(x)= [a0] * + [b0] applicable between the sections [rO] and [rl].
5. At some time during the interval between [t0] and [t1]. it is required that the temperature distribution of the rod exceeds the linear profile mu0(x) [a0] [bO] in the region from [rO] to [rl].
Table 14 shows the character-level relative edit distance between each natural language problem and its 5 augmentations based on the problem type and the number constraints.
D. Collection of Manually Written PDE Control Problems
We overview questionnaire in Fig. 10. For the questionnaire collection, each Zoom session takes hour, including background instruction and manual design (with interactive guidance). Although it is challenging to make fair statistical comparison with human experts in terms of performance (both time cost and accuracy), our LLMs (autoformalization + Controller + code generation) complete the task in under 120 seconds on a single 6000 Ada GPU. This is significantly faster than the human volunteers we recruited (for collecting our real-world samples), who required several minutes just to read the original informal problem. We further show the statistics of the background of participants in Fig. 11.
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDES
Table 14: Averaged relative edit distance between the problem's natural language and GPT augmentions. Deviations are parentheses.
Relative Edit Distance
PDE
Number of Constraints
3
Heat
0.431 (0.0545)
0.456 (0.0505)
0.490 (0.0398)
Wave
0.455 (0.0524)
0.472 (0.0433)
0.484 (0.0352)
Difference between Synthetic and Manual Data We observe some differences between synthetic and manually generated data that may lead to the model's 's failure to produce valid STI. logic:
Ambiguous symbol usage, such using "ho" instead of "rho" denote material density.
Inconsistent units within single sentence. For example, "Assume that the discretized time interval is 0.05s and the maximum time is 7400 milliseconds."
Insufficient information, where four samples fail to fully describe the material properties. For instance, the manual data only provides the density of one material after the statement "the rod composed of two materials".
1.Delies Your
2.5miie Your Mom Pritee
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Temparature
Haar Squason
Hount Source
Design PDE Control Problems
"The atrecchalkoompee
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Force F Herterd
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Figure 10: Google Form for collecting manually written PDE control problems.
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PDE- Con-roont:l LLMs for Autoformalization and Reasoning of PDES
Math
PhD (graduated or current)
50%
EE/CS
Physics
Master (graduated current)
Mechanics
43.8%
18.8%
6.3%
Others
Undergraduate
(graduated or current)
43.8%
37
Figure 11: Background of questionnaire participants.
E. Training Details
We leverage the pretrained token context length. All our trained models are evaluated zero-shot. For fair comparison of our models with MathCoder2, we provide two few-shot examples for the latter to leave sufficient tokens to generate a valid output (whether STL. or Python). We similarly provide two few-shot examples for GPT-4o and ol-mini.
Beroooder2--eeSeeeaaaieee (Lu et al., 2024) checkpoint (MathCoder2) which has a 4096-
Our instructions are structured in the Alpaca format (Taori al., 2023).
E.1. Autoformalization: SFT of Translator
The Translator was trained with two 6000Ada GPUs using a per-GPU batch size of 16 and 4 gradient accumulation steps for parameter model from (Lu al., 2024) with LoRA
total of 3000 steps. We fine-tuned the Sethhodder2-DepSeekkMee rank 64 and 256.
Prompt: Below is a natural language description of partial differential equation optimization problem. Translate the problem into Latex code following -iinnl temporal logic.
E.2. Program Synthesis: SFT of Coder
The Coder further fine-tuned the Translator with supervised fine-tuning and LoRA, rank 64 and 256, to produce Python code from natural language and STL pairs. This was trained with two 6000Ada GPUs using a per-GPU batch size of 8 and 8 gradient accumulation steps for a total of 3000 steps.
design two prompts, firstly for generating python code aligned with the anchor STL, and secondly for generating python code aligned with the proposed subgoal STL. The natural language problem is provided both cases extract system settings.
Prompt: Below is natural language description of partial differential equation optimization problem, paired with your spatial-signal temporal logic description of the same problem provided earlier. Note that there may be mistakes in the spatial-signal temporal logic statement but the natural language description is accurate. Translate the problem into Python code following spatial-signal temporal logic.
Prompt: Below natural language description of partial differential equation optimization problem, paired with your spatial-signal temporal logic description of an intermediate problem provided earlier. Instead of optimizing the natural language problem directly, want to optimize the intermediate problem to produce state that will better serve to achieve the final conditions outlined in the natural language problem. Your spatial-signal temporal logic description in latex paired to the original problem describes this intermediate problem. Translate the intermediate problem into Python code following spatial-signal temporal logic.
In practice, we find to be helpful to supervise fine-tune the Coder with misaligned (NL, STL) pairs to promote the subgoal STL constraints when synthesizing the corresponding Python program for the subgoal STL..
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
E.3. Reasoning: RLHF of Controller
The Controller is trained with DPO (Rafailov al., 2024) from the Translator checkpoint with LoRA rank r 64 and 256. We train with two do00Ada GPUs using a per-GPU batch size of 2 and 4 gradient accumulation steps for total of 16,800 steps. For DPO, we set 3 0.1, and À in Eq. 3.
Below is a natural language description of partial differential equation optimization problem. Instead of optimizing the provided problem directly, we want to optimize an intermediate problem to produce a state that will better serve to achieve the final conditions outlined in the natural language problem. Generate a spatial-signa temporal logic description in Latex code for such an intermediate problem.
Prompt:
F. More Experiment Results
F.1. End-to-End Evaluation of Program Synthesis
In Table 15 and Table 16, we provide end-to-end results of program synthesis on synthetic and manual data, respectively. In these results, coders take LLM-generated (noisy) STLs in their prompt. Overall, our Coder LLM still achieves strong results. Moreover, we also show another baseline "Coder-only"", where the Coder L.I. takes only natural language as the input without explicitly formalized STLs. In Table 16, we can see that our method (autoformalization + program synthesis) outperforms "Coder-only" (direct program synthesis without autoformalization), emphasizing the importance of leveraging formal language (STL..
Table 15: End-to-end autoformalization and program synthesis. The Coder produces Python using the Translator's STL output. MathCoder2 produces Python using its own STL output. Coder-only is MathCoder2 model fine-tuned to produce python directly from natural language and seeing no STL. To be comparable with the Translator Coder autoformalization and program synthesis, Coder-only is trained for 6000 steps with the same settings as the Translator and Coder (Appendix E). The evaluation for Coders is zero-shot. MathCoder2 is evaluated with two few-shot examples. Deviations over seeds are in parentheses. Bold indicates the best, underline indicates the runner up.
Executahility (*)
Utility
PDE
Model
(Coder)
RMSE (L)
Ours
0.9978 (0.0015)
0.0174 (0.0065)
Heat
MathCoder2
0.9197 (0.02731)
1.3841 (0. 1196)
Ours (Coder-onlyy
0.9978 (0.0009)
0.0235 (0.00163)
Ours
0,9620 (0.0098)
0,0076 (0.0011)
Wave
MathCoder2
0.8305 (0.0475)
0.8332 (0.0965)
Ours (Coder-only)
0.9779 (0.002148)
0.02522 (0.000583)
Table 16: End-to-end autoformalization and program synthesis on manually written data (Sec. 3.2). The Coder produces Python using the Translator's s STI. output. MathCoder2 produces Python using its own STL output. Coder-only is a MathCoder2 model fine-tuned to produce python directly from natural language and seeing STL. To be comparable with the Translator Coder autoformalization and program synthesis. Coder-only trained for 6000 steps with same settings as the Translator and Coder (Appendix E). The evaluation for Coders is zero-shot. MathCoder2 is evaluated with two few-shot examples. Deviations over seeds are in parentheses. Bold indicates the best, underline indicates the runner up.
Executability (1)
Utility
PDE
Model
(Coder)
RMSE (L)
Ours
0.4510 (0.0832)
0.1837 (0.0095)
Hcat
MathCoder2
0.4902 (0.1386)
0.2928 (0.0780)
Ours (Coder-only)
0.6078 (0.0555)
2.442 (0.0)
Ours
1.0 (0.0)
0.0119 (0.0)
Wave
MathCoder2
0.9020 (0.0277)
1.767 (0.8109)
Ours (Coder-only)
0.4706 (0.0)
0.0890 (0.0)
F.2 PDE Reasoning
1. END-TO-ENU EVALUATION OF PDE REASONING
We include end-to-end evaluation results, where we use the Coder LLM to generate Python code after the Controller LLM propose subgoal STLs in Table 17. In general, our Controller still shows strong reasoning capability (both the success rate
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
Table 17: End-to-end scientific reasoning over PDE control problems via our Controller LLM. Deviations over seeds are in parentheses. "Valid STL o' (%)" is the ratio of proposed subgoal STL o' without any syntax errors. Bold indicates the best, underline denotes the up.ner-up. "x" indicates no valid STLs were generated for evaluation. "." indicates not applicable.
Success Rate P
Utility Gain Art
Difficulty Level
PDE
Math-
GPT
GPT
Math-
GPT
GPT
Ours
Ours
Coderv2
mil-min)
(4o)
Coderv2
(al -mini)
(4o)
Easy
0.490 1666) 0.399 (0.2605)
0.154 (0) 0.160 (0.0686) 0.651 (1.1550) 0.352 (1.3309)
-1.262 (0)
-0.865 (2.0310)
Medium
0.345 1640) 0.339 (0.2345)
(0) 0.121 (0.0497)
1.0055 (0.9657)
-0.061 (0.4098)
-2.455 (0)
-2.294 (1.7773)
Heat
Hard
0.307 (0. 1363) 0.239 (0.1895) 0.074 (0.0262) 0.150 (0.0797)
7.788 (2.1369)
-1.254 (0.8047)
-3.992 (0.5175)
-2.603 (2.4486)
All
0.381 (0. (0.1556) 0.326 (0.2282)
0.144 (0.0660)
-0.731 (1.4059)
-0.321 (0.8485)
X
-1.921 (2.0856)
Easy
0.897 (0.0502) 0.854 (0.0894)
1 (0)
1.433 (0.4135) 1.173 (0.6424)
X
1.907 (0.0333)
Medium
0.836 (0.0936) 0.743 (0.0410)
0.958 (0.0105)
0.865 (0.4508)
0.758 (0.4265)
X 0.803 (0.2279)
Wave
Hard
0.331 (0.12144) 0.336 (0.1041)
0.416 (0.1113)
-0.289 (0.5723)
-0.649 (0.3948)
X -0.376 (0.2142)
0.688 (0.0884) 0.644 (0.0781)
0.791 (0.0406)
0.698 (0.57733
0:427 (0:4879)
x
0.778 (0.1585)
Valid STL of (%) (t)
82.70 (1.97)
42.45 (10.54)
0.04 (0.10)
2.55 (0.65)
82,70 (1.97)
42.45 (10.54)
0.04 (0.10)
2.55 (0.65)
Valid Python Code (5) (1)
75.65 (1.50)
27.95 (5.72)
0.09 (0.78)
3.25 (0.63)
75.65 (1.50)
27.95 (5.72)
0.09 (0.78)
3.25 (0.63)
Table 18: Proportion of valid subgoal STL generations by different controller models in Table and (in decimal form). Deviations over 5 seeds are in parentheses.
Proportion (E [0.
valid STT. o' (t)
Difficulty Lovel
PDE
Math-
GPT
GPT
Ours
Coderv2
(ol-mini)
(4o)
Easy
[088851 (0.0110) 0.507 (0.1354)
0
0.028 (0.0078)
Medium
0.891 (0.0243) 0.491 (0.0962) 0.0004 (0.0015) 0.031 (0.0046)
Heat
Hard
0.881 (0.0148) 0.493 (0.0956) 0.002 (0.0045)
0.052 (0.0113)
All
0.886 (0.0167) 0.497 (0.1091) 0.0008 (0.002)
0.037 (0.0079)
Easy
0.756 (0.0228)
0.309 (0.1113)
0
0.008 (0.0066)
Medium
0.748 (0.0257)
0.397 (0.0947)
0
0.015 (0.0052)
Wave
Hard
0.799 (0.0197) 0.349 (0.0990)
0 0.018 (0.0036)
0.768 (0.0227) 0.352 (0. (0.1016)
0.014 (0.0051)
Table 19: Proportion of valid subgoal Python program generations by different coder models in Table 17 (in decimal form). Deviations
over 5 seeds are in parentheses.
Proportion (E 00 1l of valid Python program (1)
Difficulty Level
PDE
Math-
GPT
GPT
Ours
Coderv2
(ol-mini)
(4o)
Easy
0.754 (0.0244) 0.301 (0.0624) 0.014 (0.0193) 0.047 (0.0113)
Medium
0.767 (0.0261) 0,295 (0.0503) 0.012 (0.0124) 0.039 (0.0048)
Heal
Hard
0.828 (0.0113) 0.329 (0.0532) 0.027 (0.0147)
0.058 (0.0085)
All
0.783 (0.0206) 0.308 (0.0553) 0.018 (0.0155)
0.048 (0.0082)
Easy
0.702 (0.0080) 0.211 (0.0675)
0
0.012 (0.0048)
Medium
0.713 (0.0113) 0.278 (0.0538)
0
0.018 (0.0041)
Wave
Hard
0.776 (0.0086) 0.265 (0.0561)
0
0.021 (0.0044)
All
0.730 (0.0093) 0.251 (0.0591)
0
0.017 (0.0044)
and utility gain), and also high rate of proposing valid subgoal STL (6))
F.2.2. PROPORTION OF VALID PREDICTIONS
We notice that the number of valid predictions varied significantly depending on the model. Therefore, we include Table 18 for STL and Table 19 for Python program to comprehensively show the number of valid predictions that each model makes under each type of problem and difficulty level. Please note that the values in Table 18 and Table 19 are not expressed as percentages.
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PDE-Controlle:: LLMs for Autoformalization and Reasoning of PDEs
G. More Related Works
G.1. Autoformalization in AI-for-math
Autoformalization the process of converting informal mathematical statements or instructions into formal representations, is explored through a variety of techniques. A significant subset of works employed LLMs to translate informal descriptions into formal representations. VernaCopter (van de Laar et al., 2024) leveraged LLMSs to convert natural language commands into Signal Temporal Logic (STL) specifications, integrating syntax and semantic checkers for correctness. Pretrained LLMs like GPT-3.5 and GPT-40 were leveraged to translate informal problems into Isabelle proof sketches, refining outputs through iterative prompting and heuristic-based validation (Zhou et al., 2024). Back-translation (Jiang et al., 2023) trained LLMs to map between informal and formal theorem statements in Lean4 and Isabelle. These approaches focused on leveraging LLMs for direct autoformalization while incorporating filtering mechanisms to improve reliability. In contrast, hybrid approaches interact between manual and autoformalization. Several studies combined expert-curated manual formalization with automated techniques to improve accuracy. DeepSeek-Prover (Xin et al., 2024), Trigo (Xiong et al., 2023), and (Murphy et al., 2024) used an iterative pipeline where initial formalization was manually crafted, followed by automated expansion and refinement. LILA (Mishra et al., 2022) similarly applied domain-specifie rules and Python-based DSL annotations for automatic formalization while relying on human annotators for cases where automation failed. These hybrid approaches aimed to balance the scalability of automation with the precision of manual verification. In our work, we for the first time train LLMs to autoformalize informal PDE control problems into formalized STL logic.
G.2. PDE Controls
PDEs are essential for modeling physical phenomena, helping researchers predict behaviors, optimize processes, and drive innovation across fields like climate modeling and material design. PDE control focuses on manipulating system behaviors, ensuring stability in applications like robotics and reactors, while enabling systems to adapt to changing conditions for more sustainable solutions (Alvarez, 2020; Holl et al., 2020; Ramos al., 2022; Mukherjee & Liu, 2023; Wei et al.; 2024b). Adjoint methods have been widely used for controlling physical systems governed by PDEs due to their accuracy, despite being computationally expensive (Lions, 1971; McNamara et al., 2004; Protas, 2008), while deep learning- based approaches, such as supervised learning with differentiable physics losses (Holl et al., 2020; Hwang et al., 2022), optimize control directly via backpropagation through time. (Mowlavi & Nabi, 2022; Barry-Straume et al., 2022) optimizes control problems and the PDE system state with physics-ifformed neural networks. Reinforcement learning (RL) (Farahmand et al., 2017: Pan et al., 2018; Rabault et al., 2019) optimizes control by treating signals as actions for sequential preision-making in fluid dynamics applications like drag reduction, heat transfer, and swimming (Garnier et al., 2021). Pretrained LLMs can enhance PDE surrogate modeling by integrating textual descriptions of system information-such as boundary conditions and the governing PDE-into into a multimodal learning framework (Lorsung & Farimani, 2025). PDE control can also be discretized and formulated into mixed-integer linear programming (MILP) problem via finite element method (FEM) (Sadraddini & Belta, 2015; Alvarez, 2020). More recently, diffusion- feffusioo--bseed generation has been leveraged to jointly optimize the PDE simulations and control signals (Wei et al., 2024a). Different from traditional PDE control objectives such as setpoint/rajectory tracking and disturbance rejection (Paunonen & Phan, 2019; Paunonen & Humaloja, 2022), our utility score objective (Appendix A.2), which mainly aims reduce distance to the target, can better handle inequality constraints compared to tracking errors. From an optimization perspective, our work leverages LLMs to propose better initializations (initial conditions) to solve the open-loop PDE control problem. Extensions to closed-loop control are possible by appending the utility from the LLM-proposed optimization into future optimization rounds. This increases the complexity of LLM fine-tuning; thus, as the first step in this direction we focus on open-loop control.
G.3. LLMe Task Planning
Natural language (NL.)based task planning in robotics has gained increasing attention. Approaches such (Pan et al., 2023) enable task-specific translations from informal language to Linear Temporal Logic (LTL), allowing robots to follow structured plans even in low-resource scenarios. Building on this foundation, recent research has explored the use of LLMs for task planning, demonstrating models' potential in Accision-making and executing complex plans (Singh et al., 2023; Shah et al., 2023; Li et al., 2022). For instance, (Wang et al.) leveraged LLMS to arrange and predict execution sequences for robots, achieving a comparable success rate to human users. Additionally, (Ren et al., 2023) addressed the hallucination issue from LLM-based planners by incorporating uncertainty alignment, improving the reliability of generated plans. More recently, CLMASP (Lin al., 2024) retined LLM-generated skeleton plans using Answer Set Programming (ASP) for robotic task execution. Inspired by these subgoal approaches, we for the first time train LLMs to perform reasoning and planning on PDE control problems.
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PDE-Controlle:: LLMs for Autoformalization and Reasoning of PDEs
H. Datasheets for Datasets
This document based on Datasheets for Datasets by (Gebru et al., 2021).
H.1. Motivation
For what purpose was the dataset created? Was there a specific task mind? Was there specific gap that needed to be filled? Please provide a description.
The dataset was created to enable large language models (LLMs) to tackle complex Partial Differential Equation (PDE) control problems. The specific purpose to advance automated formalization and reasoning in applied mathematics, addressing the lack of datasets tailored PDE-related tasks. The dataset bridges informal natural language problems and formal specificatioons/odee for PDE systems, fostering research in scientific computing and engineering.
Who created this dataset (e.g., which team, research group) and on behalf of which entity (e.g,, company, institution, organization)?
The dataset was created by the anonymous authors of this PDE-Controller paper, affiliated with a research group focused on Al-for-math applications.
What support was needed to make this dataset? (e.g. who funded the creation of the dataset? If there is an associated grant, provide the name of the grantor and the grant name and number, or if it was supported by a company or government agency, give those details.)
The creation of the dataset was supported by research funding for developing novel applications of L.LMS applied mathematics. Further support included computational resources for fine-tuning LLMs and manual curation by domain experts.
Any other comments?
The dataset represents a pioneering effort to merge Al capabilities with PDE-based scientific reasoning, significantly expanding the potential applications of LLMs.
H.2. Composition
What do the instances that comprise the dataset represent (e.g., documents, photos, people, countries)? Are there multiple types of instances (e.g., movies, users, and ratings; people and interactions between them; nodes and edges)? Please provide a description.
Each instance represents a PDE control problem, including: Informal problem descriptions in natural language; 2) Formal specifications in Signal Temporal Logic (STL); Python code that integrates PDE simulation and optimization tools.
How many instances are there in total (of each type, if appropriate)? The dataset comprises over 2.13 million synthetic (natural language, STL, Python code) triplets, with additional real-world examples including 17 manually written heat problems and 17 wave problems.
Does the dataset contain all possible instances or is it sample (not necessarily random) of instances from larger set? If the dataset is sample, then what is the larger set? Is the sample representative of the larger set (e.g., geographic coverage)? If so, please describe how this representativeness was validated/verified.. If it is not representative of the larger set, please describe why not (e.g., to cover a more diverse range of instances, because instances were withheld or unavailable).
synthesized dataset designed to cover diverse range of PDE control problems, sampled and augmented from representative templates ensure coverage of key scenarios and complexities.
What data does each instance consist of? "Raw" data (e.g., unprocessed text or images) or features? In either case, please provide a description.
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDES
Each instance includes: 1) Informal natural language descriptions of PDE problems; 2) Formal representations in STL syntax; 3) Python code for solving the PDE problem using optimizers such as Gurobi.
Is there label or target associated with each instance? If so, please provide description. Yes, each instance includes ground-truth STL and Python code, verified for alignment and executability.
Is any information missing from individual instances? so, please provide a description, explaining why this information is missing (e.g.. because it was unavailable). This does not include intentionally removed information, but might include, e.g., redacted text. Not Applicable.
Are relationships between individual instances made explicit (e.g., users' movie ratings, social network links)? so, please describe how these relationships are made explicit. Yes, relationships between natural language, STI. specifications, and Python code are explicitly maintained for traceability.
Are there recommended data splits (e.g., training, development/validattoon testing)? If please provide description of these splits, explaining the rationale behind them. Yes, the dataset is split into training and testing sets, with specific splits for heat and wave problems.
Are there any errors, sources of noise, or redundancies in the dataset? so, please provide a description. Synthetic data validated through automated checks and human verification. Errors may arise from annotation inconsistencies, especially in manually curated problems.
Is the dataset self-contained, or does it link to or otherwise rely on external resources (e.g., websites, tweets, other datasets)? If it links to or relies on external resources, a) are there guarantees that they will exist, and remain constant, over time; b) are there official archival versions of the complete dataset (i.c., including the external resources as they existed at the time the dataset was created); c) are there any restrictions (e.g., licenses, fees) associated with any of the external resources future user? Please provide descriptions of all external resources and any restrictions associated with them, well as links other access points, appropriate.
that might apply
The dataset is self-contained, with no reliance on external or dynamic resources.
Does the dataset contain data that might be considered confidential (e.g., data that is protected by legal privilege or by doctor-patien confidentiality, data that includes the content of individuals' non-public communications)? so, please provide description. No.
Does the dataset contain data that, if viewed directly, might be offensive, insulting, threatening, or might otherwise cause anxiety? If so, please describe why.
No.
Does the dataset relate people? If not, you may skip the remaining questions in this section. No.
Does the dataset identify any subpopulations (e.g., by age, gender)? If so, please describe how these subpopulations are identified and provide a description of their respective distributions within the dataset.
No.
Is it possible to identify individuals (i.e., one more natural persons), either directly indirectly (i.e., in combination with other data) from the dataset? If so, please describe how.
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PDE-Controlle:: LLMs for Autoformalization and Reasoning of PDEs
No.
Does the dataset contain data that might be considered sensitive in any way (e.g,, data that reveals racial or ethnic origins, sexual orientations, religious beliefs, political opinions or union memberships, or locations; financial or health data; biometric or genetic data; forms of government identification, such as social security numbers; criminal history)? If so, please provide a description. No.
Any other comments?
The dataset's richness in complexity and diversity makes it significant resource for advancing applied mathematics via Al.
H.3. Collection
How was the data associated with each instance acquired? Was the data directly observable (e.g., raw text, movie ratings), reported by subjects (e.g., survey responses), or indirectly inferred/derived from other data (c.g., part-of-speech tags, model-based guesses for age or language)? If data was reported by subjects or indirectly inferred/derived from other data, was the data validated/verified? If so, please describe how.
The data was synthesized from key PDE control templates, augmented through principled methods, and verified by experts. Real-world problems collected via guestionnaire-basee manual curation process involving students and researchers.
Over what timeframe was the data collected? Does this timeframe match the creation timeframe of the data associated with the instances (e.g., recent crawl of old news articles)? If not, please describe the timeframe in which the data associated with the instances was created. Finally, list when the dataset was first published. The synthetic dataset was generated in late 2024, with real-world problems curated in 2025.
What mechanisms or procedures were used to collect the data (e-g., hardware apparatus or sensor, manual human curation, software program, software API)? How were these mechanisms or procedures validated? Procedures included automated STL. generation, natural language augmentation using GPT-4, and manual problem formulation.
What was the resource cost of collecting the data? (e.g. what were the required computational resources, and the associated financial costs, and energy consumption estimate the carbon footprint.)
Resource costs included computational expenses for data synthesis and human time for manual curation and validation.
If the dataset is sample from a larger set, what was the sampling strategy (e.g., deterministic, probabilistic with specific sampling probabilities?? Not applicable.
Who was involved in the data collection process (e.g., students, crowdworkers, contractors) and how were they compensated (e.g., how much were crowdworkers paid)? Graduate students and researchers with expertise in applied mathematics and AI.
Were any ethical review processes conducted (e.g., by an institutional review board)? so, please provide a description of these review processes, including the outcomes, as well link or other access point to any supporting documentation. No.
Does the dataset relate to people? If not, you may skip the remainder of the questions in this section.
No.
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDES
Did you collect the data from the individuals in question directly, obtain it via third parties other sources (e.g., websites)?
Our manually written data is collected from each individual with questions.
Were the individuals question notified about the data collection? so, please describe (or show with screenshots or other information) how notice was provided, and provide link or other access point to, or otherwise reproduce, the exact language of the notification itself. N/A
Did the individuals in question consent to the collection and use of their data? If so, please describe (or show with screenshots or other information) how consent was requested and provided, and provide a link or other access point to, or otherwise reproduce, the exact language to which the individuals consented. N/A
If consent obtained, were the consenting individuals provided with mechanism to revoke their consent in the future or for certain uses? If so, please provide description, as well as link or other access point to the mechanism (if appropriate)
N/A
Has an analysis of the potential impact of the dataset and its use on data subjects (e-g., a data protection impact anen conducted? If so, please provide description of this analysis, including the outcomes, as well link other access point any supporting documentation. No. Our data are intended to be used in evaluation only and all charts are publicly avialable.
Any other comments? N/A
H.4. Preprocessing / Labeling
Was any prepooessingdddennnng/aelligg of the data doneee.g..isccretizaai or bucketing, tokenization, part-of- speech tagging, SIFT feature extraction, removal of instances, processing of missing values)? If so, please provide a description. If not, you may skip the remainder of the questions in this section.
Yes, preprocessing included: 1) Reformatting natural language prompts; 2) Validating STL and Python code for correctness and executability; Augmenting natural language data using rephrasing techniques.
Was the "raw" data saved addition to the proproessed/clennddhhbee data (c.g., to support unanticipated future uses)? If so, please provide link or other access point to the "raw" data.
Yes, raw data and intermediate representations are retained for reproducibility and future use.
Is the software used to preprocess/ceann/abel the instances available? If so, please provide link or other access point. The tools and scripts for preprocessing are included in the supplementary materials of the PDE-Controller framework.
Any other comments?
Preprocessing ensures high-quality alignment between natural language, formal logic, and executable code.
H.S. Uses
Has the dataset been used for any tasks already? If so, please provide description. Yes, was used train and evaluate the PDE-Controlles framework and benchmark its performance against state-of of-t----trhrrr L.Mss.
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PDE-Controlle: LLMs for Autoformalization and Reasoning of PDEs
Is there repository that links to any all papers systems that use the dataset? If so, please provide link or other access point.
N/A
What (other) tasks could the dataset be used for?
The dataset could be used for: 1) Training models for scientific reasoning and formalization; 2) Developing tools for automated program synthesis; 3) Advancing research in AI-driven engineering and physics.
Is there anything about the composition of the dataset or the way it was collected and preprocssed//deanel/lbbee that might impact future uses? For example, is there anything that a future user might need to know to avoid uses that could result in unfair treatment of individuals or groups (e.g., stereotyping, quality of service issues) or other undesirable harms (e.g., financial harms, legal risks) If so, please provide a description. Is there anything a future user could do to mitigate these undesirable harms? N/A
Are there tasks for which the dataset should not be used? If so, please provide a description. The dataset is not suitable for tasks unrelated to PDE control or tasks requiring real-world human data.
Any other comments? The dataset's structured format supports reproducible and extensible research in applied mathematics.
H.6. Distribution
Will the dataset be distributed to third parties outside of the entity (e.g., company, institution, organization) behalf of which the dataset was created? If so, please provide a description. Yes, the dataset will be made publicly available for research purposes.
How will the dataset will be distributed (e.g., tarball on website, API, GitHub)? Does the dataset have a digital object identifier (DOI)? The dataset will be distributed via GitHub and academic repositories, with accompanying documentation.
When will the dataset be distributed? The dataset is expected to be released following the ICML 2025 conference.
Will the dataset be distributed under a copyright or other intellectual property (IP) license, and/or under applicable terms of use (ToU)? If so, please describe this license and/or ToU, and provide link or other access point to, or otherwise reproduce, any relevant licensing terms or ToU, as well any fees associated with these restrictions. Yes, it will be distributed under permissive license (e.g., CC BY-SA 4.0) to encourage research use.
Have any third parties imposed IP-ed or other restrictions on the data associated with the instances? If so, please describe these restrictions, and provide a link or other access point to, or otherwise reproduce, any relevant licensing terms, as well as any fees associated with these restrictions. All charts are subjected to their respective copyrights by the authors of this paper.
Do any export controls or other regulatory restrictions apply to the dataset or to individual instances? If so, please describe these restrictions, and provide a link or other access point to, or otherwise reproduce, any supporting documentation.
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
N/A
Any other comments?
Distribution will include detailed usage guidelines to ensure proper application of the dataset.
H.7. Maintenance
Who is uuportingghostinggmanitin the dataset? The authors of the PDE-Controller framework.
How can the ovner/curatorr/maaager the dataset be contacted (e.g., email address)?
Contact information will be provided with the dataset release.
Is there an erratum? so, please provide link or other access point. http::://pdd-coot.ooo
github.io/
Will the dataset be updated (e.g., correct labeling errors, add new instances, delete instances)? so, please describe how often, by whom, and how updates will be communicated to users (e.g., mailing list, GitHub)? Yes, periodic updates will incorporate additional real-world problems and refinements.
If the dataset relates to people, are there applicable limits on the retention of the data associated with the instances (e.g., were individuals in question told that their data would be retained for fixed period time and then deleted)? so, please describe these limits and explain how they will be enforced. N/A
Will older versions of the dataset continue to be supportechosec/maaittanne?? If so, please describe how. If not, please describe how its obsolescence will be communicated to users. Yes, previous versions will remain accessible for reproducibility.
If others want to extend/augment/huild on/contribute to the dataset, is there mechanism for them do so? If so, please provide description. Will these contributions be validated/verified? If so, please describe how. If not, why not? Is there process for communicating/distibbe Yes, contributions will be encouraged through a collaborative platform (e.g., GitHub).
these contributions to other users? If so, please provide description.
Any other comments?
The dataset's maintainers are committed to ensuring its long-term usability and relevance for scientific research.
Misc.
URL to benchmark. The benchmark URL can be found here: https::/ppe-coottoo////
URL to Croissant metadata. The Croissant metadata URL can be found here: https://huugiinggacccco
datasets deeess/eelaaaaaaaa -atttllaaaaiii ai/aaa//// ai/pee-eeei-----eeee-eee
Author statement & license information. We the authors bear all responsibility in case violation of rights.
Hosting and maintenance. We have dedicated webpage for hosting instructions: httpp::/ppde-cootzooll/ gi github.ie /o/. We committed to performing major maintenance periodically.
Dataset Structure. All files are stored in the JSONL format. For the translator dataset, store separate files based on STL syntax formats, the number of constraints, and the train-test split. Each training file contains more than 600 samples, and each test file contains more than 1.50 samples. Each sample includes the informal question in natural language, the STI.
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PDE-Controller: LLMs for Autoformalization and Reasoning of PDEs
representation in LaTeX, and the corresponding Python script.
For the preference dataset, split the files based three difficulty levels and the train-test split. Each sample in the file contains the informal question in natural language, the winner STL that improves the informal question, and the loser STL that worsens it. For each STL, we also provide the resulting utility score.
32


Reviews:
[{"review_id": "jDBZZcoviJ", "reviewer": "Reviewer_kzDp", "summary": "This paper develops PDE-Controller that uses LLMs to solve open-loop control inputs for PDEs with constraints. The PDE-Controller uses LLMs to transform informal natural language instructions into formal specifications in the form of STL, and then combine optimization solvers with LLM reasoning to improve the utility of PDE control. It has been observed that the PDE-Controller significantly outperforms GPT4o and a few open-source models in utility gain for PDE control.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We deeply appreciate your feedback and suggestions.\n> Q1\n1) Our utility score (A.2) can faithfully quantify whether STL (constraints) are fully met by solutions simulated by the solver. This utility is inherited from [1] and serves as the rule-of-thumb “accuracy” metric. pp3p affirms “evaluation criteria are well-structured and appropriate for assessing the framework's performance” and Bygn: “metrics [...] are appropriate and effectively capture the nuances of PDE control scenarios”.\n2) We further explain the connection b/w our metrics and pass@k below:\n* For autoformalization (Translator), IoU is equivalent to “average@k”, it averages the alignment b/w predictions and targets over multiple generations and problems. We can discretize IoU into “pass@k” by considering only pass/fail cases based on token-level differences; but this ignores fine-grained quantifications of autoformalization. In the tables below, IoU and pass@k are not always aligned.\n* For code generation (Coder), the executability metric is essentially pass@k from the executability perspective.\n\nHeat\n|Model|IoU|Pass@1|pass@2|pass@3|\n|---|---|---|---|---|\n|Ours|0.992(0.07)|0.978(0.142)|0.980(0.134)|0.982(0.131)|\n|MathCoder2|0.772(0.35)|0.538(0.480)|0.565(0.484)|0.583(0.493)|\n\nWave\n|Model|IoU|Pass@1|pass@2|pass@3|\n|---|---|---|---|---|\n|Ours|0.992(0.07)|0.971(0.161)|0.975(0.152)|0.977(0.149)|\n|MathCoder2|0.1953(0.045)|0.3305(0.4396)|0.3726(0.4663)|0.3971(0.4893)||\n\n[1] \"Formal Methods for Partial Differential Equations\" Alvarez 2020.\n> Q2\n\nYes, vDyK affirms: “Methods and evaluation criteria are clearly defined”. Bygn: “The designs are sound, adequately controlled, and effectively validate the proposed model’s strengths.” To demonstrate that our components are essential, we clarify the ablation comparisons below based on Heat problem results (Table 3 & 10):\n* Our Translator component is essential. It has better autoformalization abilities; +28.5% IoU over MathCoder2.\n* Our Coder component is essential and robust to noisy autoformalization:\n  * Given ground truth STL, our Coder has +4% better executability rate in generated Python than MathCoder2, and utility RMSE is 91.6% lower (better) than MathCoder2.\n  * When switching from noisy Translator predictions to ground truth STL, our Coder’s utility only drops 0.57% indicating that our Coder is robust under noisy STL inputs. We will include this discussion in our camera ready.\n|Method|Performance|\n|---|---|\n|Mathcoder’s translator(Table3)|IoU=0.772|\n|Our Translator(Table3)|IoU=0.992(+28.5%)|\n|Mathcoder’s coder(Table3)|Executability=0.9592|\n|Our Coder(Table3)|Executability=0.9978(+4.02%)|\n|Mathcoder’s coder(Table3)|UtilityRMSE=0.2058|\n|Our Coder(Table3)|UtilityRMSE=0.0173(-91.6%)|\n|TranslatorSTL→Coder(Table10)|UtilityRMSE=0.0174|\n|ground truth STL→Coder(Table3)|UtilityRMSE=0.0173(-0.57%)|\n\n> Q3\n\nWe have added the o1-mini reasoning model (Table 3, 4, 6). This cost-efficient version, suitable for our academic budget, performs comparably to the o1 model in math and coding tasks. OpenAI’s official statement: “o1-mini may outperform o1-preview when it comes to coding applications” [1] [2]. Further “o1‑mini excels at STEM, especially math and coding” and “for applications that require reasoning without broad world knowledge” [3].\n\n[1] [OpenAI](https://help.openai.com/en/articles/9855712-openai-o1-models-faq-chatgpt-enterprise-and-edu)\n\n[2] [Benchmark test](https://aimlapi.com/comparisons/openais-o1-preview-vs-o1-mini)\n\n[3] [OpenAI](https://openai.com/index/openai-o1-mini-advancing-cost-efficient-reasoning/)\n\n[4] and Table 1 and Fig. 6 in Wu, S., et al. 2024. A Comparative Study on Reasoning Patterns of OpenAI's o1 Model. arXiv:2410.13639\n> Q4\n\nThank you for raising this important issue. Indeed, closed-loop control is more realistic in applications. It is among future routes we plan to explore. Extending to closed-loop control is possible by appending the utility from the subgoal optimization into future optimization rounds. But this increases the complexity of LLM fine-tuning. Thus, as the first step in this direction we focus on open-loop control. We will include this discussion and provide clarification in our camera ready. \n> Q5\n\n1) Both setpoint tracking and our solver (A.2~A.3) require discretizing PDEs and constraints and designing cost functions or utility scores to characterize PDE control objective satisfaction. The core differences lie in the cost functions’ or utility scores’ design and formulation. Our utility score can better handle inequality constraints (A.2) compared to tracking errors, which mainly aim to reduce distance to the target.\n2) Our work does not explicitly model disturbances such as (thermal) noise or variations in material properties (e.g., diffusivity).\nOverall, we see no key barriers to replacing our current solver with those for setpoint tracking or disturbance rejection.\n\n> Q6\n\nPlease see Bygn Q1 (due to character limits).\n> Q7 \n\nPlease see vDyK Q1 (due to character limits).", "claims_and_evidence": "To demonstrate the advantage of PDE-Controller, the authors got 2 million synthetic samples for the control of heat and wave equations, and gathered another 34 human-written problems. Then the authors developed various metrics such as IoU and Utility RMSE and use such metrics to support their claim.\n\nBased on their results, I am convinced that their proposed PDE-Controller framework can achieve good utility.", "methods_and_evaluation_criteria": "The metrics proposed in this paper make sense to me. There may be several other metrics that are needed. For example, it may be beneficial if the authors can include another metric that measures whether the STL are fully met by the solutions. In addition, there is a gap between the metrics in this paper and the standard pass@k metrics used in the LLM literature.\n\nOn evaluation methods, some ablation study is also missing. There are a few components for PDE-Controller. It is unclear whether all these components are that essential, and some ablation study could be helpful.\n\nFinally, it may be useful if the authors can include one reasoning model (e.g. o1) for their evaluations.", "theoretical_claims": "This paper does not have theoretical contributions.", "experimental_designs_or_analyses": "I read all the experimental results. The metrics make sense, and the results are solid. However, I have also commented on a few things that I think are missing.", "relation_to_broader_scientific_literature": "Overall, this paper is relevant to the big area of LLMs for science and engineering. However, the scope of this paper is confined to a very specific question: how to generate open-loop control for heat or wave equations with up to three STLs. The scope is quite narrow. Clearly, the paper will be significantly improved if they can i) consider closed-loop control, ii) consider more complicated PDEs (e.g, 2D NS equation), iii) scale for more than three STLs. I am mostly concerned regarding the first item. Due to the uncertainty of real systems, sensing, actuation, and feedback are typically needed to deploy closed-loop PDE control. This paper completely ignores this issue. It is also unclear to me how the utility and STLs used in the formulation of this paper are connected to the traditional PDE control objectives such as setpoint/trajectory tracking and disturbance rejection.", "essential_references_not_discussed": "There is a large body of textbooks and papers on closed-loop PDE control, which are not mentioned by this paper. If the authors do not want to touch on closed-loop PDE control, maybe it is worth revising the paper to emphasize this at the beginning of the paper. I mean, from what I understand, when control people talk about \"PDE control\", they typically mean \"closed-loop PDE control.\"", "other_strengths_and_weaknesses": "The way how this paper uses LLMs for PDE control is original. The significance is questionable since the paper does not consider closed-loop control, more complicated PDEs, as well as the case with more than three STLs.", "other_comments_or_suggestions": "The paper is well written and I have not noticed many typos.", "questions_for_authors": "1. It may be beneficial if the authors can include another metric that measures whether the STL are fully met by the solutions. In addition, is there a way to connect the evaluations with the standard pass@k metrics used in the LLM literature?\n\n2. Have the authors considered doing a comprehensive ablation study?\n\n3. Have the authors considered adding a one reasoning model (e.g. o1) as a strong baseline?\n\n4. Closed-loop PDE control is typically preferred for real systems. Have the authors considered closed-loop control?\n\n5. How are the utility and STLs used in the formulation of this paper connected to the traditional PDE control objectives such as setpoint/trajectory tracking and disturbance rejection? \n\n6. Can the authors extend their method for more complicated PDEs such as 2D NS equations?\n\n7. Can the authors comment on the possibility of scaling their method for more than three STLs?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kGjdFL1PMs", "reviewer": "Reviewer_pp3p", "summary": "The paper proposes that the PDE-Controller framework enhances large language models (LLMs) to control systems governed by PDEs, addressing their limitations in rigorous logical reasoning. It transforms natural language instructions into formal specifications, improving PDE control's reasoning, planning, and utility. The holistic solution includes datasets, math-reasoning models, and novel metrics, outperforming existing models by up to 62\\% in utility gain. This work bridges language generation and PDE systems, showcasing LLMs' potential in scientific and engineering applications.", "questions": "", "limitations": "", "rating": 5, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We deeply appreciate your feedback and suggestions.\n\n> Have you explored alternative reinforcement learning algorithms besides DPO for training the controller? Given recent advancements in reasoning-enhanced LLMs, comparing the performance of GRPO or other RL-based methods in training the PDE controller would be valuable.\n\nFor alternative RL algorithms besides DPO, we also experimented with Eq. 3 without the SFT regularization term and found that it led to degraded generation and overfitting. Our Eq. 3 is stable and achieves strong performance. We will include this discussion in the camera ready.\nWe agree that further study of GRPO [1] and the latest RL methods is valuable, and we plan to explore them in our Controller training in future work.\n\n[1] “DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning” DeepSeek-AI (Arxiv 2501.12948)", "claims_and_evidence": "The claims presented in the paper are well-supported by empirical evidence.", "methods_and_evaluation_criteria": "The proposed methodology and evaluation criteria are well-structured and appropriate for assessing the framework's performance.", "theoretical_claims": "No explicit theoretical claims are presented in the paper.", "experimental_designs_or_analyses": "The experimental design is well-justified, supporting the claims regarding the effectiveness of LLMs in PDE control. The authors thoroughly analyze the framework's impact on reasoning and control performance.", "relation_to_broader_scientific_literature": "This research presents a novel contribution at the intersection of LLMs and applied mathematics, particularly in PDE control. This area has received limited attention in the literature.", "essential_references_not_discussed": "The paper adequately discusses relevant prior work.", "other_strengths_and_weaknesses": "1) Novel framework for PDE control automation: The paper introduces an innovative approach that leverages LLMs for reasoning-based PDE control.\n\n2) New dataset: The dataset enables evaluating LLMs' reasoning capabilities in PDE control scenarios.\n\n3) Significant performance improvements: The framework outperforms existing models considerably.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1) Have you explored alternative reinforcement learning algorithms besides DPO for training the controller? Given recent advancements in reasoning-enhanced LLMs, comparing the performance of GRPO or other RL-based methods in training the PDE controller would be valuable.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "DnsfnAMxHP", "reviewer": "Reviewer_bygn", "summary": "The paper introduces PDE-Controller, a framework leveraging large language models (LLMs) for automating the formalization and reasoning of control problems governed by partial differential equations (PDEs). The authors claim significant performance improvements in translating informal natural language PDE control problems into formal specifications using Signal Temporal Logic (STL), synthesizing executable Python code, and proposing effective intermediate reasoning subgoals. Experimental results demonstrate up to 62% improvement in PDE control utility over baseline LLM models, supported by a newly created dataset comprising over 2 million samples.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We deeply appreciate your feedback and suggestions.\n\n> The paper sufficiently addresses related works but could further discuss recent developments in differentiable physics and physics-informed neural networks (PINNs) which also address PDE control.\n\nThank you for the suggestion! We will include a discussion on recent developments in differentiable physics and physics-informed neural networks (PINNs) in the related work section [1, 2]. It will be an interesting study to replace our numerical solver with neural operators in our framework.\n\n[1] “Learning to control pdes with differentiable physics”\n\n[2] “Solving pde-constrained control problems using operator learning”\n\n\n> Q1. How would your method scale to multi-dimensional PDE problems?\n\nExtending our method to multi-dimensional PDEs, such as 2D Navier Stokes, will involve enhancing the Gurobi solver, which is a topic for our future work. Nonetheless, the development of more advanced solvers will not impact our core contributions to the design and fine-tuning of our LLMs.\n\n\n> Q2. Have you evaluated or planned internal optimization methods to reduce dependency on external solvers like Gurobi?\n\nWe indeed plan to develop internal optimizers for more general PDE settings, to reduce our dependence on external solvers.\nMeanwhile, we may not completely eliminate the use of external solvers. Integrating well-developed external solvers into LLMs can be a strength in solving complex problems, as demonstrated in the following examples:\n1) In theorem proving, any LLMs depend on their proof environments to construct proofs. [1] integrates their LLM based prover with the Lean proof environment to present promising premises for interactive proof generation. [2] enables LLMs that integrate with Lean for tactic suggestion, proof search, and premise selection. [3] and [4] are further works that require Lean.\n2) Moreover [5] presents an LLM framework that autoformalizes natural language linear programming problems then calls an external optimizer for optimization.\n\n\n[1] Yang, K., Swope, A. M., Gu, A., Chalamala, R., Song, P., Yu, S., Godil, S., Prenger, R., & Anandkumar, A. (2023). LeanDojo: Theorem Proving with Retrieval-Augmented Language Models. arXiv preprint arXiv:2306.15626.\n\n[2] Song, P., Yang, K., & Anandkumar, A. (2025). Lean Copilot: Large Language Models as Copilots for Theorem Proving in Lean. arXiv preprint arXiv:2404.12534\n\n[3] Wang, R., Zhang, J., Jia, Y., Pan, R., Diao, S., Pi, R., & Zhang, T. (2024). TheoremLlama: Transforming General-Purpose LLMs into Lean4 Experts. arXiv preprint arXiv:2407.03203.\n\n[4] Lin, Y., Tang, S., Lyu, B., Wu, J., Lin, H., Yang, K., Li, J., Xia, M., Chen, D., Arora, S., & Jin, C. (2025). Goedel-Prover: A Frontier Model for Open-Source Automated Theorem Proving. arXiv preprint arXiv:2502.07640. \n\n[5]  Zhang, J., Wang, W., Guo, S., Wang, L., Lin, F., Yang, C., & Yin, W. (2024). Solving General Natural-Language-Description Optimization Problems with Large Language Models. arXiv preprint arXiv:2407.07924\n\n\n\n\n> Q3. How does your approach handle poorly formulated or noisy natural language inputs in real-world scenarios?\n\nCurrently, we add robustness in our LLM against noises in real-world scenarios via natural language data augmentation with chatGPT. To improve our future research, we plan to introduce more structure-level augmentations to our synthetic data for additional robustness against noisy inputs. We fine-tune our models on synthetic data mainly because it is time and resource consuming to manually curate data. It took 17 human-hours to collect 17 heat and 17 wave problems.", "claims_and_evidence": "- Claim: PDE-Controller significantly outperforms baseline models in PDE control reasoning.\n  - Evidence: Demonstrated 62% improvement in utility gain compared to GPT-4o and MathCoder2.\n\n- Claim: PDE-Controller effectively formalizes informal PDE problems into STL and Python code.\n  - Evidence: Achieves autoformalization accuracy of over 64% and program synthesis accuracy over 82%.\n\n- Claim: The Controller model effectively decomposes complex PDE problems into manageable subgoals.\n  - Evidence: Empirical results show higher success rates and substantial improvements in utility using subgoal decomposition compared to random and baseline models.", "methods_and_evaluation_criteria": "The proposed methods and evaluation criteria, including autoformalization, subgoal reasoning via reinforcement learning from human feedback, and synthesis of executable Python programs, are appropriately tailored for PDE control tasks. The benchmarks and metrics (IoU, executability, utility RMSE) are appropriate and effectively capture the nuances of PDE control scenarios.", "theoretical_claims": "The paper does not explicitly focus on new theoretical proofs but instead emphasizes methodological innovations and empirical validations. Hence, there are no direct theoretical proofs to evaluate.", "experimental_designs_or_analyses": "Experiments were conducted on 1D heat and wave PDE problems. The authors performed comprehensive evaluations using various metrics (IoU, executability, utility RMSE), comparing PDE-Controller against established baselines (MathCoder2, GPT-4o). The designs are sound, adequately controlled, and effectively validate the proposed model’s strengths.", "relation_to_broader_scientific_literature": "The contributions relate closely to recent advances in AI-for-math and PDE control literature, particularly highlighting the gap between general-purpose LLMs and specialized scientific reasoning capabilities. The use of STL for formalization and the combination of reinforcement learning with human feedback align well with contemporary approaches in both AI-for-science and formal methods.", "essential_references_not_discussed": "The paper sufficiently addresses related works but could further discuss recent developments in differentiable physics and physics-informed neural networks (PINNs) which also address PDE control.", "other_strengths_and_weaknesses": "Strengths:\n\n- Innovative use of LLMs for formalization and reasoning in PDE control.\n\nWeaknesses:\n\n- Dependence on external optimization solvers (e.g., Gurobi) limits standalone applicability.\n\n- Limited exploration beyond 1D problems.", "other_comments_or_suggestions": "Consider additional comparisons with differentiable physics or physics-informed neural networks for completeness.", "questions_for_authors": "- How would your method scale to multi-dimensional PDE problems?\n\n- Have you evaluated or planned internal optimization methods to reduce dependency on external solvers like Gurobi?\n\n- How does your approach handle poorly formulated or noisy natural language inputs in real-world scenarios?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FDj2VU1Pnb", "reviewer": "Reviewer_vDyK", "summary": "This paper presents PDE-Controller, a framework that enables large language models (LLMs) to automate the control of systems governed by partial differential equations (PDEs).\n\nThe study highlights the gap between current AI-for-math research, which excels in pure mathematical reasoning, and its limited application in applied mathematics, particularly PDEs. The authors propose a novel pipeline that integrates autoformalization, scientific reasoning, and program synthesis for PDE control. The PDE-Controller translates informal natural language problem descriptions into formal specifications, executes reasoning steps to improve control efficiency, and generates executable code.\n\nThe model is trained using supervised fine-tuning (SFT) and reinforcement learning from human feedback (RLHF), leveraging a large dataset of synthetic and human-annotated PDE problems. Experimental results demonstrate that PDE-Controller achieves significant improvements over baseline LLMs in utility gain, autoformalization accuracy, and program synthesis efficiency.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We deeply appreciate your feedback and suggestions.\n\n> 1. The experimental design relies heavily on synthetic data generation from a limited set of template rules. This may pose challenges when generalizing to different formats.\n\nWe fine-tune our models on synthetic data mainly because it is time and resource consuming to manually curate data. It took 17 human-hours to collect 17 heat and 17 wave problems. We will try our best to collect more manually written samples in the next step.\nIt is possible to scale our method to more than three STLs; there are no additional technical challenges. We limited our dataset to three constraints to balance between exhaustive coverage of the logic formulations with computation constraints: in total we spent about two months on preparing our dataset, including collecting 2 million samples, data augmentations, and PDE simulations.\n\nDue to limited time during the rebuttal, below we show the test performance on generalizing to unseen 4-constraint STLs of our Translator (IoU) and Coder (executability & utility), over 5 problems each for heat and wave problems. Note that both models have never seen problems with 4 STLs. MathCoder2 is evaluated with 2-shot in-context examples of 4 constraints giving it an advantage. We will include these results in our camera-ready.\n\n|PDE|Model|IoU (Translator)|Executability($\\uparrow$) (Coder)|UtilityRMSE($\\downarrow$)|\n|----------|----------|----------|----------|----------|\n|Heat|Ours|0.934 (0.0)|0.8 (0.0)|0.0 (0.0)|\n|Heat|MathCoder2|0.8154 (0.0)|0.6 (0.0)|0.2600 (0.0)|\n|Wave|Ours|1.0 (0.0)|0.8 (0.0)|0.1515 (0.0)|\n|Wave|MathCoder2|0.9690 (0.0)|0.8 (0.0)|0.2393 (0.2268)|\n\n> 2. The comparison to established (non-learning) methods is lacking. It is unclear how much this approach improves beyond existing human-centric methods, whether in terms of labor reduction or accuracy.\n\nDue to the time and resource demands of having individual human experts manually formulate a given problem, code and optimizable problem and potentially reason about subgoals, collecting a large number of human-centric solutions is difficult. We will try our best to include this comparison in the camera-ready version.\n\n\n> 3. Might worth reviewing and maybe mentioning: Explain Like I'm Five: Using LLMs to Improve PDE Surrogate Models with Text (arXiv preprint arXiv:2410.01137) author: Lorsung, Cooper and Farimani, Amir Barati\n\nThank you for your suggestion. We will add this to our related work.\n\n> Other Comments Or Suggestions:\n> * Line 216 (Left) – pairs -> triplets\n> * Line 317 (Left) – and and\n> * Figure 6 – the meaning of A, B and C (i.e., the constraints?) are not explained\n\nThank you for your comments. We will correct and clarify these for the camera ready.", "claims_and_evidence": "- Results are robust and convincing.\n\n- Claims are supported by the results", "methods_and_evaluation_criteria": "Methods and evaluation criteria are clearly defined. They seem to be comprehensive and complete.", "theoretical_claims": "There are no clear theoretical claims. The work is mainly focused on adapting known methods and approaches to a new, unexplored domain.", "experimental_designs_or_analyses": "- The experimental design relies heavily on synthetic data generation from a limited set of template rules. This may pose challenges when generalizing to different formats.\n\n- The comparison to established (non-learning) methods is lacking. It is unclear how much this approach improves beyond existing human-centric methods, whether in terms of labor reduction or accuracy.", "relation_to_broader_scientific_literature": "The work utilizes state-of-the-art methods and models, demonstrating clear advantages when applying these advanced techniques to new domains.", "essential_references_not_discussed": "Might worth reviewing and maybe mentioning: \nExplain Like I'm Five: Using LLMs to Improve PDE Surrogate Models with Text (arXiv preprint arXiv:2410.01137)\nauthor: Lorsung, Cooper and Farimani, Amir Barati", "other_strengths_and_weaknesses": "- **Strengths:**  \n  i. The authors explicitly address LLM reasoning by introducing sub-goal generation and optimization, which were not originally present, and compare the results achieved with and without these components.  \n  ii. The study leverages both SFT and RLHF to improve performance when utilizing sub-goals.  \n  iii. The paper is well-written, with a clear and concise presentation that is easy to read and follow.\n\n- **Weaknesses:**  \n  i. *(Lines 430-431)* Failures on real manual data highlight the proposed method's limitations in generalizing to real-world cases and its reliance on synthetic data generated from a limited set of templates. However, the fact that other models also fail suggests that the proposed method has merits, even if its capabilities remain constrained.", "other_comments_or_suggestions": "- Line 216 (Left) – pairs -> triplets\n\n- Line 317 (Left) – and and\n\n- Figure 6 – the meaning of A, B and C (i.e., the constraints?) are not explained", "questions_for_authors": "No further questions.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper introduces "PDE-Controller," a novel framework leveraging Large Language Models (LLMs) for the challenging task of formalizing, reasoning about, and controlling Partial Differential Equations (PDEs)-governed systems. The authors propose a pipeline involving natural language understanding, translation to formal specifications (STL), subgoal reasoning, and code generation, supported by new datasets and evaluation metrics. The work demonstrates significant performance improvements over strong baselines like GPT-4o on heat and wave equation control problems.

The paper received generally positive reviews, with three out of four reviewers recommending acceptance. Reviewers broadly agreed on the paper's strengths, including the novelty of applying LLMs to PDE control automation (pp3p, bygn, vDyK), the creation of valuable datasets (human and large-scale synthetic) (pp3p), the significant empirical performance improvements shown (pp3p, bygn), and the well-structured methodology and evaluation (pp3p, bygn, vDyK). The work is seen as a valuable contribution at the intersection of LLMs and applied mathematics/scientific computing.

Reviewer kzDp raised concerns regarding the scope (limited to open-loop control, specific 1D PDEs, few constraints), the connection to traditional control objectives, and the need for specific metrics (pass@k, STL satisfaction) and ablation studies. The authors provided a detailed and rebuttal addressing these concerns. They clarified how their utility metric relates to STL satisfaction and pass@k, pointed to existing results for ablation evidence, added a requested baseline (o1-mini), justified the open-loop focus as a first step towards closed-loop control, discussed scalability (including preliminary positive results on 4-constraint problems), and addressed the role of external solvers and data limitations. It looks like the concerns are addressed, despite that reviewer didn't have a chance to update the review post-rebuttal.

Based on the majority support from the reviewers, the convincing and detailed author rebuttal that addressed the raised concerns, i recommend accepting this paper.
Decision: Accept (poster)