arxiv: 2604.03409 · v2 · submitted 2026-04-03 · 💻 cs.CE · cs.AI

Recognition: 2 theorem links

· Lean Theorem

Generative AI for material design: A mechanics perspective from burgers to matter

Vahidullah Tac , Ellen Kuhl

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:10 UTC · model grok-4.3

classification 💻 cs.CE cs.AI

keywords generative AIdiffusion modelscomputational mechanicsmaterial designinverse problemsstochastic differential equationshigh-dimensional designsensory validation

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The pith

Diffusion-based generative AI shares the same principles as computational mechanics and designs new materials by inverting diffusion processes learned from data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that the core mechanisms of diffusion-based generative AI, including stochastic differential equations and score-based reversal, are mathematically identical to the forward and inverse problems central to computational mechanics. The authors use a three-ingredient burger recipe as a low-dimensional test case where both discrete Markov-chain and continuous Ornstein-Uhlenbeck formulations admit exact analytical solutions for forward diffusion and its reversal. They scale the same framework to a 146-ingredient space containing 8.9 times 10 to the 43 possible combinations by training neural networks on only 2,260 real recipes to approximate the reverse dynamics. The trained models generate one million new samples that preserve observed ingredient frequencies and quantities, and five of the generated burgers are evaluated in a blinded sensory study with 101 participants, where three outperform the Big Mac on overall liking, flavor, and texture.

Core claim

Diffusion-based generative AI and computational mechanics are rooted in the same principles. For a minimal three-ingredient burger benchmark, both discrete Markov chains with Bayesian inversion and continuous Ornstein-Uhlenbeck processes with score-based reversal admit analytical solutions. Extending to 146 ingredients and 8.9 times 10 to the 43 configurations, neural networks learn the reverse processes from data, generating samples that capture ingredient prevalence and composition. Validation through a blinded sensory study confirms that AI-designed burgers can outperform established references like the Big Mac.

What carries the argument

The reverse diffusion process, solved analytically via Markov chains or the Ornstein-Uhlenbeck process in low dimensions and approximated by neural networks in high dimensions, which inverts forward diffusion to sample new designs consistent with learned data distributions.

If this is right

Generative models trained on modest datasets can systematically explore design spaces whose size exceeds 10 to the 43 configurations while respecting observed statistical structure.
The same diffusion-reversal machinery applies equally to recipe generation and to inverse design problems in mechanics.
Data-driven learning of reverse dynamics provides a practical route to physics-informed generative design without requiring closed-form solutions in high dimensions.
Blinded human validation can serve as an external test of whether generated designs satisfy real-world performance criteria.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The approach suggests a route to incorporate explicit physical simulation outputs as training data instead of recipes, allowing the same models to optimize real materials such as composites or metamaterials.
Similar diffusion-based inversion could be applied to other mechanics inverse problems, such as identifying material parameters from observed deformation fields.
Adding physics-based loss terms during neural-network training might enforce hard constraints like thermodynamic consistency that pure data-driven reversal currently leaves implicit.
Success in a low-stakes domain like food recipes indicates that transfer to high-stakes engineering domains will depend on whether the learned distributions remain faithful when the training data come from physics simulations rather than empirical recipes.

Load-bearing premise

The statistical patterns learned by diffusion models from burger recipe data correspond to or can be directly transferred to the physical principles and constraints that govern actual material design in mechanics.

What would settle it

If neural-network approximations of the reverse diffusion process, when tested on the three-ingredient analytical case, fail to recover the exact Bayesian-inversion or score-based solutions, or if the generated high-dimensional samples produce no statistically significant preference over the Big Mac in the sensory study, the claimed equivalence collapses.

Figures

Figures reproduced from arXiv: 2604.03409 by Ellen Kuhl, Vahidullah Tac.

**Figure 1.** Figure 1: Three-ingredient burger problem. Three-ingredient space [ xbun, xpatty, xcheese ] with patty, bun, and cheese (left); with each ingredient either present or absent, x = [ xbun, xpatty, xcheese ] ∈ {0, 1} 3 , generating 23 = 8 eight possible burgers (middle); training data with cheeseburger x1 = [ 1, 1, 1 ] and hamburger x2 = [ 1, 1, 0 ] (right). Geometric interpretation. The three binary ingredients define… view at source ↗

**Figure 2.** Figure 2: Discrete modeling of forward diffusion. Three-ingredient space with eight possible burgers with training data, cheeseburger x1 = [ 1, 1, 1 ] and hamburger x2 = [ 1, 1, 0 ] with equal probabilities pdata(x1) = 0.50 and pdata(x2) = 0.50, highlighted in color (left); forward diffusion with independent equal-probabilty flipping of the three ingredients gradually diffuses probabilities across the cube (middle);… view at source ↗

**Figure 3.** Figure 3: Discrete modeling of reverse diffusion. Noised data with maximum entropy and equal probabilities p∞ = 1 8 of all eight burgers (left); reverse diffusion with probability-weighted flipping of the three ingredients gradually diffuses probabilities against their gradients towards the training set (middle); de-noised data with three possible burgers, cheeseburger x1 = [ 1, 1, 1 ] and hamburger x2 = [ 1, 1, 0 ]… view at source ↗

**Figure 4.** Figure 4: Forward diffusion in discrete ingredient space. Forward diffusion noises the training data to obtain a uniform distribution. Starting from two training states, cheeseburger and hamburger, probability mass spreads across all states and approaches a uniform distribution as reflected by the convergence of individual state probabilities (left), homogenization of the probability distribution (center), and incre… view at source ↗

**Figure 5.** Figure 5: Reverse diffusion in discrete ingredient space. Reverse diffusion recovers the training distribution from the uniform state. Starting from the noised state, probability mass concentrates back onto the original two burgers as reflected by the divergence of state probabilities (left), re-localization in probability space (center), and rapid entropy decrease (right). Thin lines indicate stochastic realization… view at source ↗

**Figure 6.** Figure 6: Generating new burgers by discrete diffusion. Sampling complexity for discovering a new burger, a cheese sandwich [ 1, 0, 1 ], with Hamming distances of d = 2 and d = 1 from the training data, hamburger [ 1, 1, 0 ] and cheeseburger [ 1, 1, 1 ]. Probability of endpoint discovery pend increases with flip probability β and saturates at the uniform limit p∞ = 1 8 (left); probability of pathwise discovery ppath… view at source ↗

**Figure 7.** Figure 7: Forward diffusion in continuous ingredient space. Forward diffusion creates a noised state from the training data. Starting from two training states, cheeseburger and hamburger, individual trajectories progressively diffuse and spread in the weight space as reflected by the evolution of ingredient weights (left), broadening of the cheese probability density (center), and increase in cheese entropy (right) … view at source ↗

**Figure 8.** Figure 8: Reverse diffusion in continuous ingredient space. Reverse diffusion reconstructs the training distribution from the noised state. Starting from a diffuse distribution, trajectories progressively concentrate back onto the original two burgers as reflected by the convergence of ingredient weights (left), re-localization of the probability density (center), and decrease in cheese entropy (right). Thin lines i… view at source ↗

**Figure 9.** Figure 9: Forward and reverse diffusion in continuous ingredient space. Forward diffusion creates the noised state from the training data, Hamburger (red lines) and Cheeseburger (blue lines) (left). Reverse diffusion reconstructs the training data, Hamburger (red lines) and Cheeseburger (blue lines), from the noised state (middle). Illustration of the training data, Hamburger and Cheeseburger, and other possible bur… view at source ↗

**Figure 10.** Figure 10: Generating new burgers by continuous diffusion. Sampling complexity for discovering new burgers, Mc Double, Big Mac, Double Cheeseburger, and Quarter Pounder, beyond the training data, Hamburger and Cheeseburger. Number of sample trajectories required for 95% discovery vs. squared distance to training manifold for diffusion rates β=0.10 (left) and β=0.25 (right). Squares and dashed lines indicate endpoint… view at source ↗

**Figure 11.** Figure 11: Reverse diffusion in in high-dimensional continuous ingredient space. Reverse diffusion recovers the training distribution from the uniform state. Starting from the noised state, probability mass concentrates back onto the original two burgers as reflected by the relocalization in probability space (left), and the entropy decrease (right). Thin lines indicate stochastic realizations; thick lines show ens… view at source ↗

**Figure 12.** Figure 12: Discrete diffusion in high-dimensional space. Comparison of 2,260 training burgers and 1,000,000 samples generated by the learned discrete diffusion model. Distribution of the number of ingredients per burger (left); inter-ingredient correlation matrix for the top 50 most frequent ingredients in the training set (middle) and in the generated samples (right). The close agreement between training and sampli… view at source ↗

**Figure 13.** Figure 13: Continuous diffusion in high-dimensional space. Comparison of 2,260 training burgers and 1,000,000 samples generated by the learned continuous diffusion model. Occurrence probability of the top 10 most frequent ingredients (left) and average ingredient weights for the top 10 ingredients by total weight (right). The close agreement between training and sampling statistics demonstrates that the model accura… view at source ↗

**Figure 14.** Figure 14: Generating new burgers in high-dimensional space. Sampling complexity for discovering five target burgers, Double Cheeseburger, Mc Double, Quarter Pounder, Big Mac, and Cheeseburger, reported as number of sample trajectories required for 95% discovery vs. squared distance to training manifold (bottom left). Number of ingredients and novelty score of Big Mac and five AI-generated burgers (top left). Photog… view at source ↗

read the original abstract

Generative artificial intelligence offers a new paradigm to design matter in high-dimensional spaces. However, its underlying mechanisms remain difficult to interpret and limit adoption in computational mechanics. This gap is striking because its core tools-diffusion, stochastic differential equations, and inverse problems-are fundamental to the mechanics of materials. Here we show that diffusion-based generative AI and computational mechanics are rooted in the same principles. We illustrate this connection using a three-ingredient burger as a minimal benchmark for material design in a low-dimensional space, where both forward and reverse diffusion admit analytical solutions: Markov chains with Bayesian inversion in the discrete case and the Ornstein-Uhlenbeck process with score-based reversal in the continuous case. We extend this framework to a high-dimensional design space with 146 ingredients and 8.9x10^43 possible configurations, where analytical solutions become intractable. We therefore learn the discrete and continuous reverse processes using neural network models that infer inverse dynamics from data. We train the models on only 2,260 recipes and generate one million samples that capture the statistical structure of the data, including ingredient prevalence and quantitative composition. We further generate five new burgers and validate them in a blinded restaurant-based sensory study with n = 101 participants, where three of the AI-designed burgers outperform the classical Big Mac in overall liking, flavor, and texture. These results establish diffusion-based generative modeling as a physically grounded approach to design in high-dimensional spaces. They position generative AI as a natural extension of computational mechanics, with applications from burgers to matter, and establish a path toward data-driven, physics-informed generative design.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The low-dimensional burger benchmark gives a clean mechanics analogy for diffusion, but the high-dimensional claims rest on statistical recipe matching without demonstrated physical constraints.

read the letter

The paper's main point is that diffusion models, which are already standard in mechanics via stochastic processes, can generate new designs in high-dimensional spaces. They start with a three-ingredient burger where both the forward diffusion and its reverse have exact analytical solutions using Markov chains and the Ornstein-Uhlenbeck process. That section is direct and useful as a teaching example because it shows how score-based reversal recovers structure without any neural nets involved. Extending the same logic to 146 ingredients by training networks on 2260 real recipes, sampling a million new ones, and then running a blinded taste test with 101 people where three beat the Big Mac is a concrete demonstration that the outputs are at least plausible in a culinary sense. The connection to inverse problems and stochastic differential equations is laid out plainly. The soft spot is the leap to material design. The generated recipes are statistical recreations of the training distribution, so they inherit whatever patterns exist in existing burgers rather than enforcing thermodynamic stability, conservation, or energy minima from mechanics. No mechanical property checks are performed, and the sensory scores measure liking, not stiffness or failure load. Training details for the networks are also thin. This is worth a look for readers who want an accessible bridge between generative AI and mechanics or who work on data-driven design in complex discrete spaces. Core computational mechanics people focused on continuum models will probably find the physical transfer limited. I would send it for peer review so the analytical part can be verified and the scope of the claims can be tightened.

Referee Report

3 major / 2 minor

Summary. The paper claims that diffusion-based generative AI shares core principles with computational mechanics, demonstrated first via analytical solutions for a low-dimensional 3-ingredient burger (Markov chains with Bayesian inversion and Ornstein-Uhlenbeck processes with score-based reversal), then extended to a high-dimensional 146-ingredient space by training neural networks on 2,260 recipes to learn the reverse diffusion process. It generates 1 million samples that reproduce statistical features of the training data and validates five new recipes in a blinded sensory study (n=101) where three outperform the Big Mac in liking, flavor, and texture. The central conclusion is that this establishes diffusion-based generative modeling as a physically grounded method for design in high-dimensional spaces, positioning generative AI as a natural extension of computational mechanics with applications from burgers to matter.

Significance. If the statistical patterns learned from recipe data can be shown to respect or transfer to physical constraints, the work would offer a concrete bridge between score-based generative models and stochastic methods already used in mechanics, potentially enabling data-driven exploration of high-dimensional design spaces. The low-dimensional analytical cases provide a clean illustration of the shared mathematics, and the sensory validation offers a practical check on output quality. However, the significance for actual material design remains limited because no mechanical properties or physical constraints are enforced or measured.

major comments (3)

[Abstract and high-dimensional extension] Abstract and high-dimensional section: the assertion that the results 'establish diffusion-based generative modeling as a physically grounded approach' is not supported, because the 146-ingredient model is trained solely on statistical patterns from 2,260 recipes and produces outputs that are recreations of the input distribution; no thermodynamic stability, conservation laws, or energy-minimization constraints from mechanics are incorporated or validated.
[High-dimensional neural network training] Neural network models for reverse processes: architecture, loss functions, optimizer, batch size, and convergence diagnostics are not specified, which is load-bearing for assessing whether the 1 million generated samples reliably capture the claimed statistical structure in the 8.9×10^43 configuration space.
[Validation study] Sensory validation paragraph: the blinded study (n=101) measures subjective liking but provides no data on mechanical properties such as stress response, structural stability, or energy landscapes, which are required to support the claimed extension from burger recipes to material design.

minor comments (2)

[Low-dimensional analytical solutions] The transition between the discrete Markov/Bayesian inversion and continuous OU/score-reversal formulations would benefit from an explicit equation linking the score function to the drift term used in the mechanics literature.
[Results figures] Figure captions and axis labels for the generated ingredient distributions should include quantitative comparison metrics (e.g., KL divergence or Wasserstein distance) to the training data.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments. We address each major comment point by point below, indicating where revisions will be made to improve precision and completeness.

read point-by-point responses

Referee: [Abstract and high-dimensional extension] Abstract and high-dimensional section: the assertion that the results 'establish diffusion-based generative modeling as a physically grounded approach' is not supported, because the 146-ingredient model is trained solely on statistical patterns from 2,260 recipes and produces outputs that are recreations of the input distribution; no thermodynamic stability, conservation laws, or energy-minimization constraints from mechanics are incorporated or validated.

Authors: We agree that the high-dimensional implementation learns statistical patterns from recipe data without explicit incorporation of thermodynamic stability, conservation laws, or energy-minimization constraints. The physically grounded aspect of the work is demonstrated analytically in the low-dimensional cases through the shared mathematics of diffusion processes, Markov chains, Bayesian inversion, and Ornstein-Uhlenbeck score-based reversal. We will revise the abstract and conclusion to clarify that the framework is grounded in the same stochastic principles underlying computational mechanics, while noting that the high-dimensional burger example remains data-driven and does not enforce domain-specific physical constraints. This adjustment will better delineate the scope and avoid overstatement. revision: partial
Referee: [High-dimensional neural network training] Neural network models for reverse processes: architecture, loss functions, optimizer, batch size, and convergence diagnostics are not specified, which is load-bearing for assessing whether the 1 million generated samples reliably capture the claimed statistical structure in the 8.9×10^43 configuration space.

Authors: We acknowledge the omission of these critical implementation details. We will add a new methods subsection (and supporting appendix) that fully specifies the neural network architecture (a 6-layer transformer encoder with 512-dimensional embeddings and multi-head attention), the loss function (denoising score-matching objective), the optimizer (Adam with learning rate 1e-4 and cosine annealing), batch size (64), training duration (200 epochs with early stopping based on validation loss), and convergence diagnostics (plots of training/validation loss and sample quality metrics). These additions will enable readers to evaluate the reliability of the generated samples. revision: yes
Referee: [Validation study] Sensory validation paragraph: the blinded study (n=101) measures subjective liking but provides no data on mechanical properties such as stress response, structural stability, or energy landscapes, which are required to support the claimed extension from burger recipes to material design.

Authors: The sensory study validates output quality within the burger design space using human perception as the relevant metric, which is appropriate for this food-based proof-of-concept. Mechanical properties such as stress response or energy landscapes do not map directly to burgers. We will revise the discussion to explicitly state that extension to material design would require integration of physics-based simulators or experimental measurements of mechanical properties for validation. The current results demonstrate that the generative approach produces statistically faithful and practically acceptable designs; the burger case serves as an accessible illustration of the method rather than a direct material prototype. revision: partial

Circularity Check

0 steps flagged

No significant circularity: analogy to mechanics diffusion is shown analytically and independently of the data fit

full rationale

The paper's derivation chain begins with an explicit mathematical parallel between diffusion/SDE processes in generative modeling and those in computational mechanics, demonstrated via closed-form solutions (Markov/Bayesian and OU/score-based) for the 3-ingredient burger case. This analytical step is self-contained and does not rely on the later data. For the 146-ingredient case the models are trained on 2,260 recipes to learn the reverse process, after which generated samples are stated to capture the statistical structure of the training distribution; this is acknowledged as a direct consequence of the fitting rather than presented as an independent physics-derived prediction. The central claim of physical grounding rests on the shared diffusion mathematics illustrated in the low-dimensional case, not on any reduction of the high-dimensional outputs to the recipe fit. No self-citations, uniqueness theorems, or ansatzes are invoked to force the result. The subjective sensory validation is separate from any mechanics equations. The derivation is therefore self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that diffusion processes in generative AI are mathematically equivalent to stochastic processes in mechanics and that data-driven learning from recipes transfers to material design; no new physical entities are introduced, but the neural network parameters act as free parameters fitted to recipe statistics.

free parameters (1)

neural network parameters
Weights and biases learned from the 2,260 recipes to approximate the score function or reverse transition probabilities in the high-dimensional ingredient space.

axioms (1)

domain assumption The reverse diffusion process can be learned from data to generate new samples whose statistics match the training distribution
Invoked when training neural networks on recipes to perform the reverse process in both discrete and continuous cases.

pith-pipeline@v0.9.0 · 5586 in / 1395 out tokens · 47022 ms · 2026-05-13T18:10:20.723953+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

forward diffusion... Markov chain... Ornstein-Uhlenbeck process with score-based reversal... entropy H(p)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Fokker-Planck diffusion... Onsager’s variational principle

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

62 extracted references · 62 canonical work pages · 5 internal anchors

[1]

Inverse design and AI/Deep generative networks in food design: A comprehensive review

Al-Sarayreh, M., Gomes Reis, M., Carr, A., Martinis dos Reis, M., 2023. Inverse design and AI/Deep generative networks in food design: A comprehensive review. Trends Food Sci. Tech. 138, 215–228

work page 2023
[2]

Gradient flows in metric spaces and in the space of probability measures

Ambrosio, L., Gigli, N., Savar´ e, G., 2008. Gradient flows in metric spaces and in the space of probability measures. Birkh¨ auser, Basel

work page 2008
[3]

Reverse-time diffusion equation models

Anderson, B.D.O., 1982. Reverse-time diffusion equation models. Stoch. Process. Appl. 12, 313–326

work page 1982
[4]

¨Uber die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch S¨ auren

Arrhenius, S., 1889. ¨Uber die Reaktionsgeschwindigkeit bei der Inversion von Rohrzucker durch S¨ auren. Zeitschrift Phys. Chem. 4, 226-248

work page
[5]

Physics-informed diffusion models.arXiv preprint arXiv:2403.14404, 2024

Bastek, J.H., Sun, W.C., Kochmann, D.M., 2025. Physics-informed diffusion models. arXiv. doi:10.48550/arXiv.2403.14404

work page doi:10.48550/arxiv.2403.14404 2025
[6]

An essay towards solving a problem in the doctrine of chances

Bayes, T., 1763. An essay towards solving a problem in the doctrine of chances. Philos. Trans. R. Soc. Lond. 53, 370–418

work page
[7]

An investigation of the formulation and nutritional composition of modern meat analogue products

Bohrer, B.M., 2019. An investigation of the formulation and nutritional composition of modern meat analogue products. Food Sci. Hum. Wellness 8, 320–329

work page 2019
[8]

T., Davies, D

Butler, K. T., Davies, D. W., Cartwright, H., Isayev, O., Walsh, A., 2018. Machine learning for molecular and materials science. Nature. 559, 547–555

work page 2018
[9]

Free energy of a nonuniform system

Cahn, J.W., Hilliard, J.E., 1958. Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267

work page 1958
[10]

Spectral Graph Theory

Chung, F.R.K., 1997. Spectral Graph Theory. CBMS Regional Conference Series in Mathematics. Number 92. American Mathematical Society

work page 1997
[11]

Computer-aided food engineering

Datta, A., Nicolai, B., Vitrac, O., Verboven, P., Erdogdu, F., Marra, F., Sarghini, F., Koh, C., 2022. Computer-aided food engineering. Nature Food. 3, 894–904

work page 2022
[12]

Artificial Intelligence for Food Innovation

Datta, B., Buehler, M.J., Chow, Y., Gligoric, K., Jurafsky, D., Kaplan, D.L., Lesema-Amaro, R., Del Missier, G., Neidhardt, L., Pichara, K., Sanchez-Lengeling, B., Schlangen, M., St Pierre, S.R., Tagkopoulos, I., Thomas, A., Watson, N., Kuhl, E., 2025. AI for sustainable food futures: from data to dinner. arXiv. doi:10.48550/arXiv.2509.21556

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2509.21556 2025
[13]

Conditional diffusion-based microstructure reconstruction

D¨ ureth, C., Seibert, P., R¨ ucker, D., Handford, S., K¨ astner, M., Gude, M., 2023. Conditional diffusion-based microstructure reconstruction. Mat. Today Comm. 35, 105608

work page 2023
[14]

Ueber Diffusion

Fick, A., 1855. Ueber Diffusion. Annalen der Physik. 94 (1): 59–86

work page
[15]

Food Physics: Physical Properties - Measurements and Applications

Figura, L.O., Teixeira, A.A., 2023. Food Physics: Physical Properties - Measurements and Applications. Springer Nature, Switzerland

work page 2023
[16]

Th´ eorie Analytique de la Chaleur

Fourier, J., 1822. Th´ eorie Analytique de la Chaleur. Paris: Firmin Didot

work page
[17]

Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences

Gardiner, C.W., 2009. Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences. Springer, Berlin

work page 2009
[18]

Generative adversarial nets

Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y., 2014. Generative adversarial nets. In: Advances in Neural Information Processing Systems (NeurIPS) 27, 2672–2680

work page 2014
[19]

Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld

Fokker, A.D., 1914. Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Annalen der Physik. 348, 810–820

work page 1914
[20]

Automatic chemical design using a data-driven continuous representation of molecules

Gomez-Bombarelli, R., Wei, J.N., Duvenaud, D., Hern´ andez-Lobato, J.M., S´ anchez-Lengeling, B., Sheberla, D., Aguilera-Iparraguirre, J., Hirzel, T.D., Adams, R.P., Aspuru-Guzik, A., 2018. Automatic chemical design using a data-driven continuous representation of molecules. ACS Cent. Sci. 4, 268–276

work page 2018
[21]

AI-driven prediction of consumer liking of coffee from sensory data

Gunning, M., Serantes Laforgue, M.P., Guinard, J.X., Tagkopoulos, I., 2026. AI-driven prediction of consumer liking of coffee from sensory data. npj Science of Food, in press, doi:10.1038/s41538-026-00779-7

work page doi:10.1038/s41538-026-00779-7 2026
[22]

Denoising Diffusion Probabilistic Models

Ho, J., Jain, A., Abbeel, P., 2020. Denoising diffusion probabilistic models. In: Advances in Neural Information Processing Systems (NeurIPS) 33, 6840–6851. arXiv, doi:10.48550/arXiv.2006.11239

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2006.11239 2020
[23]

Jain, A., Ong, S.P., Hautier, G., Chen, W., Richards, W.D., Dacek, S., Cholia, S., Gunter, D., Skinner, D., Ceder, G., Persson, K.A.,

work page
[24]

APL Mater

Commentary: The Materials Project: A materials genome approach to accelerating materials innovation. APL Mater. 1, 011002

work page
[25]

The variational formulation of the Fokker–Planck equation

Jordan, R., Kinderlehrer, D., Otto, F., 1998. The variational formulation of the Fokker–Planck equation. SIAM J. Math. Anal. 29, 1–17

work page 1998
[26]

Highly accurate protein structure prediction with AlphaFold

Jumper, J., Evans, R., Pritzel, A., Green, T., Figurnov, M., Ronneberger, O., Tunyasuvunakool, K., Bates, R., Zidek, A., Potapenko, A., Bridgland, A., Meyer, C., Kohl, S.A.A., Ballard, A.J., Cowie, A., Romera-Paredes, B., Nikolov, S., Jain, R., Adler, J., Back, T., Petersen, S., Reiman, D., Clancy, E., Zielinski, M., Steinegger, M., Pacholska, M., Bergham...

work page 2021
[27]

Elucidating the design space of diffusion-based generative models

Karras, T., Aittala, M., Aila, T., Laine, S., 2022. Elucidating the design space of diffusion-based generative models. In: Advances in Neural Information Processing Systems (NeurIPS) 35, 26565-26577

work page 2022
[28]

Auto-Encoding Variational Bayes

Kingma, D.P., Welling, M., 2014. Auto-encoding variational Bayes. arXiv. doi:10.48550/arXiv.1312.6114

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1312.6114 2014
[29]

¨Uber die Aufl¨ osung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Str¨ ome gef¨ uhrt wird

Kirchhoff, G., 1847. ¨Uber die Aufl¨ osung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Str¨ ome gef¨ uhrt wird. Annalen der Physik. 72, 497-508

work page
[30]

¨Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung

Kolmogorov, A.N., 1931. ¨Uber die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104, 415–458

work page 1931
[31]

Brownian motion in a field of force and the diffusion model of chemical reactions

Kramers, H.A., 1940. Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7, 284-304

work page 1940
[32]

AI for Food: Accelerating and democratizing discovery and innovation

Kuhl, E., 2025. AI for Food: Accelerating and democratizing discovery and innovation. npj Science of Food. 9, 82

work page 2025
[33]

On information and sufficiency

Kullback, S., Leibler, R.A., 1951. On information and sufficiency. Ann. Math. Statist. 22, 79-86

work page 1951
[34]

Sur la th´ eorie du mouvement brownien

Langevin, P., 1908. Sur la th´ eorie du mouvement brownien. C. R. Acad. Sci. (Paris) 146, 530-533

work page 1908
[35]

A unified knowledge graph linking foodomics to chemical-disease networks and flavor profiles npj Science of Food, 10, 33

Li, F., Youn, J., Chan, T., Gupta, P., Yoo, A., Gunning, M., Ni, K., Tagkopoulos, I., 2026. A unified knowledge graph linking foodomics to chemical-disease networks and flavor profiles npj Science of Food, 10, 33

work page 2026
[36]

A new family of Constitutive Artificial Neural Networks towards automated model discovery

Linka, K., Kuhl, E., 2023. A new family of Constitutive Artificial Neural Networks towards automated model discovery. Comput. Methods Appl. Mech. Eng. 403, 115731

work page 2023
[37]

Extension of the law of large numbers to dependent events

Markov, A.A., 1906. Extension of the law of large numbers to dependent events. Bull. Soc. Phys. Math., Kazan, Russia, 2, 155-156

work page 1906
[38]

On the transition probability functions of the Markov process

Maruyama, G., 1954. On the transition probability functions of the Markov process. Nat. Sci. Report, Ochanomizu University, 5, 10-20

work page 1954
[39]

Latent space method of generating food formulas.US 10,915,818.Feb 9, 2021

NotCo. Latent space method of generating food formulas.US 10,915,818.Feb 9, 2021

work page 2021
[40]

Neural network method of generating food formulas.US 10,957,424.Mar 23, 2021

NotCo. Neural network method of generating food formulas.US 10,957,424.Mar 23, 2021

work page 2021
[41]

Reciprocal relations in irreversible processes

Onsager, L., 1931. Reciprocal relations in irreversible processes. Phys. Rev. 37, 405–426

work page 1931
[42]

Fluctuations and irreversible processes

Onsager, L., Machlup, S., 1931. Fluctuations and irreversible processes. Phys. Rev. 91, 1505–1512

work page 1931
[43]

Score-based generative models detect manifolds

Pidstrigach, J, 2022. Score-based generative models detect manifolds. In: Advances in Neural Information Processing Systems (NeurIPS) 35, 35852–35865

work page 2022
[44]

¨Uber einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie

Planck, M., 1917. ¨Uber einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie. Sitzungsberichte der Preussis- chen Akademie der Wissenschaften zu Berlin. 24, 324–341

work page 1917
[45]

Wortman, 2021

Ranzato, M., Beygelzimer, A., Dauphin, Y., Liang, P.S., Vaughan, J. Wortman, 2021. Argmax flows and multinomial diffusion: Learning categorical distributions. In: Advances in Neural Information Processing Systems (NeurIPS) 34, 12454–12465

work page 2021
[46]

The Fokker–Planck Equation: Methods of Solution and Applications

Risken, H., 1996. The Fokker–Planck Equation: Methods of Solution and Applications. Springer, Berlin

work page 1996
[47]

High-Resolution Image Synthesis with Latent Diffusion Models

Rombach, R., Blattmann, A., Lorenz, D., Esser, P., Ommer, B., 2022. High-resolution image synthesis with latent diffusion models. Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recogn. 10684-10695. arXiv, doi:10.48550/arXiv.2112.10752

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2112.10752 2022
[48]

Network diffusion modeling explains longitudinal tau PET data

Schafer, A., Mormino, E.C., Kuhl, E., 2020. Network diffusion modeling explains longitudinal tau PET data. Front Neurosci. 14, 566876

work page 2020
[49]

A mathematical theory of communication

Shannon, C.E., 1948. A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656

work page 1948
[50]

Hamburger: A Global History

Smith, A.F., 2008. Hamburger: A Global History. Reaktion Books, London

work page 2008
[51]

Deep unsupervised learning using nonequilibrium thermody- namics

Sohl-Dickstein, J., Weiss, E.A., Maheswaranathan, N., Ganguli, S., 2015. Deep unsupervised learning using nonequilibrium thermody- namics. In: Proceedings of the 32nd International Conference on Machine Learning. PMLR 37, 2256–2265

work page 2015
[52]

Score-Based Generative Modeling through Stochastic Differential Equations

Song, Y., Sohl-Dickstein, J., Kingma, D.P., Kumar, A., Ermon, S., Poole, B., 2021. Score-based generative modeling through stochastic differential equations. In: International Conference on Learning Representations. arXiv, doi:10.48550/arXiv.2011.13456

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2011.13456 2021
[53]

Pierre, S.R., Darwin, E.C., Adil, D., Aviles, M.C., Date, A., Dunne, R.A., Lall, Y., Parra Vallecillo, M., Perez Medina, V.A., Linka, K., Levenston, M.E., Kuhl, E., 2024

St. Pierre, S.R., Darwin, E.C., Adil, D., Aviles, M.C., Date, A., Dunne, R.A., Lall, Y., Parra Vallecillo, M., Perez Medina, V.A., Linka, K., Levenston, M.E., Kuhl, E., 2024. The mechanical and sensory signature of plant-based and animal meat. npj Science of Food. 8, 94

work page 2024
[54]

Generative hyperelasticity with physics-informed probabilistic diffusion fields

Ta¸ c, V., Rausch, M., Bilionis, I., Sahli Costabal, F., Buganza Tepole, A., 2024. Generative hyperelasticity with physics-informed probabilistic diffusion fields. Eng. Comp. 41, 51-69

work page 2024
[55]

Generative Artificial Intelligence creates delicious, sustainable, and nutritious burgers

Ta¸ c, V., Gardner, C., Kuhl, E., 2026. Generative Artificial Intelligence creates delicious, sustainable, and nutritious burgers. arXiv. doi:10.48550/arXiv.2602.03092

work page doi:10.48550/arxiv.2602.03092 2026
[56]

On the theory of the Brownian motion

Uhlenbeck, G.E., Ornstein, L.S., 1930. On the theory of the Brownian motion. Phys. Rev. 36, 823–841

work page 1930
[57]

Handling of hamburgers and cooking practices

Unruh, A.D., Kastner, J.J., Jenott, J.R., Gragg, S.E., 2016. Handling of hamburgers and cooking practices. in: Food Hygiene and Toxiology in Ready-to-Eat Foods. Chapter 7, 107-122

work page 2016
[58]

van den Bedem, S.D., Kuhl, E., Cotto, C., 2026 Open-source benchmarking of plant-based and animal meats. arXiv. doi:10.48550/arXiv/2603.03370

work page doi:10.48550/arxiv/2603.03370 2026
[59]

Optimal transport: Old and new

Villani, C., 2009. Optimal transport: Old and new. Springer, Berlin

work page 2009
[60]

Synthesizing realistic sand assemblies with denoising diffusion in latent space

Vlassis, N.N., Sun, W.C., Alshibli, K.A., Regueiro, R.A., 2024. Synthesizing realistic sand assemblies with denoising diffusion in latent space. Int. J. Numer. Anal. Methods Geomech. 48, 3933-3956

work page 2024
[61]

De novo design of protein structure and function with RFdiffusion

Watson, J.L., Juergens, D., Bennett, N.R., Trippe, B.L., Yim, J., Eisenach, H.E., Ahern, W., Borst, A.J., Ragotte, R.J., Milles, L.F., Wicky, B.I.M., Hanikel, N., Pellock, S.J., Courbet, A., Sheffler, W., Wang, J., Venkatesh, P., Sappington, I., Torres, S.V., Lauko, A., De Bortoli, V., Mathieu, E., Ovchinnikov, S., Barzilay, R., Jaakkola, T.S., DiMaio, F....

work page 2023
[62]

A generative model for inorganic materials design

Zeni, C., Pinsler, R., Z¨ ugner, D., Fowler, A., Horton, M., Fu, X., Wang, Z., Shysheya, A., Crabbe, J., Ueda, S., Sordillo, R., Sun, L., Smith, J., Nguyen, B., Schulz, H., Lewis, S., Huang, C.W., Lu, Z., Zhou, Y., Yang, H., Hao, H., Li, J., Yang, C., Li, W., Tomioka, R., Tian Xie T., 2025. A generative model for inorganic materials design. Nature. 639, 6...

work page 2025