A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations

Mohammad Kohandel; Shalev Manor

REVIEW 4 major objections 4 minor 17 references

A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Two observed states sharing one coordinate but differing in a known factor make the unknown constant and function of a differential equation uniquely recoverable, with bounded error when the states only nearly match.

desk verdict The exact two-point identifiability result is correct, but the approximate theorem that carries the paper's practical claims is false as stated; worth a careful revision, not acceptance. read the letter →

arxiv 2505.16996 v1 pith:T6I56S5C submitted 2025-05-22 cs.LG

Shalev Manor , Mohammad Kohandel This is my paper

classification cs.LG MSC 34A5568T07

keywords physics-informedneuralnetworksuniversalinverseproblemsidentifiabilitynon-uniquenessdifferentialequationsfunctiondiscoveryparameter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that a class of inverse problems—simultaneously recovering an unknown constant and an unknown function inside a differential equation—becomes uniquely solvable once the data contains two observations that share the value of one coordinate $y$ while differing in a known state-dependent factor $C(x)$. The authors prove that for systems whose $q$-th equation has the form $\dot{x}_q = \beta g(y) + C(x)u(y) + d(x)$, such a pair of points determines $\beta$ and $u(y)$ exactly, and a second theorem shows the same cancellation works when the growth term $g(y)$ is also unknown. They further prove that when the two $y$-values are only $D$-close rather than equal, recovery still succeeds within error intervals that shrink to zero as $D\to 0$, assuming Lipschitz continuity. The practical payoff is that experimental design, not merely more data or larger networks, can eliminate the non-uniqueness that has limited simultaneous parameter and function discovery, and the paper demonstrates this on chemotherapy and Lotka-Volterra models.

What carries the argument

The carrying mechanism is the two-point cancellation identity: for states $x_1, x_2$ with a common $y$, subtracting the $q$-th components gives $(\dot{x}_1 - \dot{x}_2)_q = (C(x_1) - C(x_2))u(y) + d(x_1) - d(x_2)$, eliminating $\beta$ and expressing $u(y)$ directly from data; substituting that expression back through either state's equation yields $\beta = ((\dot{x}_1)_q - C(x_1)u(y) - d(x_1))/g(y)$. The same subtraction, repeated with $D$-close rather than equal $y$-values and a Lipschitz bound on $u$ (and possibly $g$), produces the interval bounds of Theorems 3.5 and 3.7.

What would settle it

Take a concrete instance of the Theorem 3.1 system with known Lipschitz constant $L$, a known separation $D$, and specified derivative values, then compute the interval claimed in Remark 3.6 and check whether the true $\beta$ falls inside it; because the theorem guarantees $\beta$ lies in that interval, any instance where the true value falls outside falsifies the approximate-uniqueness claim. Since the interval bound is derived in the appendix, an independent recomputation of that bound on a single numerical example is enough to run the test.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 3.1: in a system whose $q$-th component reads $\dot{x}_q = \beta g(y) + C(x)u(y) + d(x)$, with $g$, $C$, $d$, and the map $y = H_1(x)$ known, if two states $x_1, x_2$ satisfy $H_1(x_1) = H_1(x_2)$, $C(x_1) \neq C(x_2)$, and $g(y) \neq 0$, then the unknown constant $\beta$ and the unknown function $u(y)$ are uniquely determined from the two derivative values. The proof is a two-step algebraic cancellation: subtracting the component equations at the two states isolates $u(y)$ purely in terms of known quantities, and back-substitution into either equation isolates $\beta$; applying the same step at any other observed state with $C(x) \neq 0$ then recovers $u(y)$ across the sampled domain. Theorem 3.3 extends the cancellation to the case where the growth term is completely unknown, identifying both $g(Y)$ and $u(Y)$ at each shared $Y$. Theorems 3.5 and 3.7 relax the exact-equality requirement: if $u$ (and possibly $g$) is Lipschitz and the two $y$-values are within distance $D$, the recovered quantities are confined to intervals whose radii vanish as $D \to 0$.

Load-bearing premise

The load-bearing premise is that the time-derivatives $\dot{x}_i$ at the sampled states are known exactly; real data provides only the states themselves, and the neural-network experiments approximate the derivatives by autodifferentiation with no term in the error bounds accounting for that approximation.

Editorial extensions

If this is right

In the chemotherapy model introduced by Podina et al. (2024), the uniqueness conditions remove the need for two-phase data collection: a single trajectory with drug present lets a network recover $\beta$ and $u(N)$ uniquely, as Sections 4.1 and 5.1 report.
For the Lotka-Volterra system, applying Theorem 3.5 independently to each component lets a UPINN recover the constants $\alpha$, $\gamma$ and the functional terms $\beta x$, $\delta y$ from one oscillatory trajectory, with predicted values within fractions of a percent of the truth.
When no two sampled $y$-values are exactly equal, the D-close theorems still guarantee recovery to within error intervals shrinking to zero as sampling density increases, provided the unknown functions are Lipschitz.
The reformulation preserves UPINN robustness to noise and data sparsity: parameter errors stay below about 4.5% under 30% proportional noise on the Lotka-Volterra trajectory, and the chemotherapy $\beta$ recovery stays near 0.03% error down to 64 training points before degrading sharply at 8 and 4 points.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: because the cancellation identity only needs a known coefficient that differs between two equal-$y$ observations, the same proof should extend to separable terms beyond the specific $C(x)u(y)$ product, such as $C(x)$ being any known smooth function with distinct values on the level set of $y$.
Editorial inference: the error bounds of Theorems 3.5 and 3.7 grow with $D$, the Lipschitz constants, and the inverse of $|C(x_1)-C(x_2)|$, so they double as an experimental-design recipe: maximize the spread of $C(x)$ across observations and seek trajectories that revisit the same $y$ with different $C$.
Editorial inference: the proofs assume exact derivatives, so a natural next step the paper does not take is to add an explicit derivative-estimation error term to the interval bounds, or to solve for the derivative field jointly with the parameters; that would let the guarantees extend directly to noisy time-series data.
Editorial inference: the identifiability condition is checkable in advance from the sampling design (whether equal-$y$ pairs exist), which suggests embedding the condition into an active-learning loop that proposes new observations specifically to create the required matched pairs.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know about arXiv:2505.16996. The exact identifiability theorems (3.1 and 3.3) are correct, elementary, and cleanly stated, and the exact-derivative experiments in Sections 4.1-4.3 do show accurate recovery when the assumptions are met. But the approximate companion, Theorem 3.5, which is the only stated bridge from exact identifiability to real noisy data, is false as stated.

Here is a concrete counterexample. Take d=0, u≡3 (so L=0), y_i=0, y_j=0.05, C(x_i)=2, C(x_j)=1, g(y_i)=1, g(y_j)=0.5, and observed derivatives 7 and 3.5. All hypotheses of Theorem 3.5 hold with D=0.05, yet the two scalar equations for (β,u) are proportional: β+2u=7 is the same information as 0.5β+u=3.5. The solution set is a line, so no finite interval for β exists. The proof's denominator g(y_i)(C(x_i)−C(x_j))−(C(x_i)−C(x_j)) is exactly the determinant g(y_i)C(x_j)−g(y_j)C(x_i) (up to sign), and it vanishes for this instance. The theorem needs a non-degeneracy condition on that determinant, and the displayed bound is not a bound.

The appendix has further problems. Remark A.2 defines µβ := β − Correction using the true unknown β, so |β−µβ| ≤ R is true by construction and gives no computable prediction interval. Theorem 3.7's proof has sign slips in the bounding steps; it may be repairable, but the derivation as written does not rigorously establish the stated inequality.

What the paper does well is the exact two-point argument. Knowing that two observations with the same projection y but different C(x) pin down β and u(y) is a genuinely useful design rule, and the first three experiments support it. The later UPINN studies (noise, dataset size) are anecdotal: no baseline comparison against the original UPINN or the phase-based scheme, no seeds or error bars, no released code.

The paper overclaims a "unified framework" with rigorous proofs. The approximate theorems are load-bearing for the practical claims, and they are unsupported as written. I would not cite this in current form. But the exact-identifiability core is sound and worth salvaging; if the authors fix Theorem 3.5 with a determinant condition and correct error bounds, the paper could become useful. That makes it worth a serious referee's time, with the expectation of major revision. I'd bring the counterexample to a reading group, but mostly as a cautionary tale about how easy it is to miss a rank condition.

Referee Report

4 major / 4 minor

Summary. The paper studies simultaneous identification of a scalar parameter β and an unknown function u(y) in ODE models of the form ˙x_q = βg(y) + C(x)u(y) + d(x), together with a variant where the growth term g(y) is also unknown. The main theoretical results are sufficient conditions for unique identifiability: Theorem 3.1 shows that if two samples have identical y-values but different C(x) and g(y)≠0, then β and u at the sampled y-values are uniquely determined; Theorem 3.3 does the same when g(y) is unknown. Theorems 3.5 and 3.7 extend these to nearly equal y-values under Lipschitz assumptions, claiming error intervals whose radii depend on the separation D. The paper validates the conditions on three ODE examples with exact derivatives and two UPINN experiments with noisy data or varying dataset size.

Significance. The exact identifiability conditions in Theorems 3.1 and 3.3 are simple, checkable, and correctly proved, and they could be genuinely useful for experimental design in PINN/UPINN inverse problems. If a correct approximate version existed, it would be an important bridge to real data. However, the approximate theorem and its error bounds are the main practical contribution, and they are not correct as stated. The paper therefore does not currently provide a sound quantitative guarantee for near-duplicate samples, and the practical claims in Sections 5.4 and 5.5 go beyond what is established. The exact part is a modest but valid contribution; the approximate part requires substantial reworking.

major comments (4)

[Section 3.2, Theorem 3.5] Theorem 3.5 is false as stated because it omits a non-degeneracy condition on the pair (i,j). Take d=0, u≡3 (so any Lipschitz constant L=0 works), y_i=0, y_j=0.05, C(x_i)=2, C(x_j)=1, g(y_i)=1, g(y_j)=0.5, and observed qth derivatives 7 and 3.5. All hypotheses in Theorem 3.5 are satisfied with D=0.05, but the two equations are β+2u=7 and 0.5β+u=3.5, which are linearly dependent; β can be any real number with u=(7-β)/2. The set of possible β is unbounded, so no interval whose radius is controlled by D (let alone shrinking to 0 as D→0 when L=0) can contain β. The missing assumption is C(x_i)g(y_j) ≠ C(x_j)g(y_i), in addition to C(x_i)≠C(x_j) and g(y_i)≠0. Because Theorem 3.5 is the only stated bridge from exact identifiability to the noisy and unevenly sampled UPINN experiments, this is a load-bearing error.
[Appendix A, proof of Theorem 3.5] The closed-form expression for β in the proof is algebraically incorrect. Solving the two exact equations with u(y_i)=u(y_j) gives β = (C(x_i) ˙x_j - C(x_j) ˙x_i)/(C(x_i)g(y_j)-C(x_j)g(y_i)), whereas the manuscript displays the denominator g(y_i)(C(x_i)-C(x_j))-(C(x_i)-C(x_j)) = (C(x_i)-C(x_j))(g(y_i)-1). The printed denominator is wrong in its structure and can vanish in identifiable settings; for example, with g(y)=y, y_i=1, y_j=1/2, C(x_i)=1, C(x_j)=2, the printed denominator is 0 while C(x_i)g(y_j)-C(x_j)g(y_i) = -3/2 and β is identifiable from consistent data. This algebra error is intimately connected to the missing determinant condition identified above and explains why the erroneous theorem was not detected.
[Remark 3.6 / Appendix Remark A.2] The error bounds are tautological and do not describe a computable estimator. The bound |β-¯β| ≤ ... is obtained by defining ¯β := β - Correction, where Correction is an unspecified quantity depending on the true β; with this definition the inequality holds by construction and says nothing about the error of any estimator computed from data. Similarly, the 'centers of the possible intervals of existence' cannot be located from data, because the interval for u(y_i) derived in the proof is parameterized by the unknown β. A correct approximate identifiability statement must instead characterize the set of all (β,u) consistent with the data, the Lipschitz bound, and ∥y_i-y_j∥≤D, and then bound the diameter of that set; the manuscript does not do this.
[Sections 4.4-4.5] The UPINN experiments do not actually test Theorem 3.5. The theorem assumes the values ˙x_q at the sampled states are known exactly, and Section 4 states that this is true only for the experiments in Sections 4.1-4.3. In Sections 4.4 and 4.5 the derivatives are obtained by automatic differentiation of a trained surrogate, and no term in the error bounds accounts for this approximation error or for observation noise. The experiments also do not identify the specific pair (i,j) used, the separation D, or the Lipschitz constant L, so the reported errors cannot be compared with the (already incorrect) bounds. The numerical results are therefore at best anecdotal evidence for the framework.

minor comments (4)

[Section 3.1, Theorem 3.1] The statement that u(y) is uniquely determined 'for all y ∈ R such that C(y) ≠ 0' overstates the result; the proof identifies u(y) only at observed states, and C is a function of x, not of y. It should be rephrased as 'for every observed y=H1(x) with C(x)≠0'.
[Sections 4.1 and 5.3] The exact-derivative experiments are said to test Theorem 3.5, but the relevant statement is the exact Theorem 3.1; Theorem 3.5 is an approximate extension and is not needed in those settings.
[Appendix A, proof of Theorem 3.7] The sentence 'We will assume that C(x_j)>0. If it is not, then it flips the inequalities, which has no effect because we find an interval for β' refers to the wrong variable and the wrong sign; the relevant division is by C(x_i)-C(x_j), not by C(x_j), and the theorem concerns g and u rather than β.
[Throughout] There are numerous typos and spacing errors, including 'trails' for 'trials' in Sections 4.5 and 5.4, 'leaser' for 'lesser' in Section 6, and a missing space in 'constant,C represents' in Section 2.1. A careful proofreading pass is needed.

Circularity Check

2 steps flagged · score 6.0 of 10

Approximate-error analysis is circular: the interval for u(y_i) is parameterized by the same β it is used to bound, and the stated point estimate/error bound is introduced as β minus the assumed Correction rather than as a data-derived prediction.

self definitional [Appendix A, Remark A.2 (proof of Theorem 3.5)]
"If we define ¯β := β − Correction we get that |β − ¯β| = |Correction| ... Thus |β − ¯β| ≤ | C(xj)C(xi)LD / (g(yi)(C(xi) − C(xj)) − (C(xi) − C(xj))) |"

The point estimate ¯β is not obtained from data independently of the unknown β; it is introduced as β minus the very Correction whose assumed interval is then quoted as the error bound. Consequently |β−¯β|≤R is an algebraic restatement of the assumed Correction interval, not a computable prediction interval. No procedure is given for selecting ¯β from the observed derivatives and known functions without already knowing β, so the advertised error certificate is vacuous as a prediction guarantee.
self definitional [Appendix A, proof of Theorem 3.5 (derivation of the interval for u(y_i))]
"Rearranging yields: ( ˙xi − ˙xj)q − β(g(yi) − g(yj)) − C(xj)LD / (C(xi) − C(xj)) ≤ u(yi) ≤ ( ˙xi − ˙xj)q − β(g(yi) − g(yj)) + C(xj)LD / (C(xi) − C(xj)) ... Now if we plug in our found range for u(yi), we get that: β = ... + Correction"

The displayed interval for u(y_i) has β in its endpoints, so it is a family of possible u-intervals indexed by the unknown β. The proof then substitutes that β-dependent interval into β = (( ˙x_i)_q − C(x_i)u(y_i))/g(y_i) and reports β as a data-only expression plus a bounded Correction. This is not an elimination: it assumes β in order to form the u-interval, then recovers a relation involving the same β. The resulting bound on β is therefore a fixed-point tautology rather than a bound derived from the two observations.

full rationale

The exact identifiability results, Theorems 3.1 and 3.3, are not circular: they subtract two observed derivative equations, solve for the unknown function value at the coincident y, then substitute back for the constant and for general y. Those steps are self-contained and do not reuse the conclusion. The circularity is confined to the approximate bridge, Theorem 3.5 and its Appendix proof, which is the stated theoretical basis for the realistic UPINN experiments. There, the interval for u(y_i) is written in terms of β, the very quantity it is later used to determine, and Remark A.2 'predicts' ¯β by subtracting the assumed Correction from the true β, making the error inequality true by construction rather than by data-driven estimation. I also note separately that Theorem 3.5 is false as stated because it omits a determinant condition, so two D-close observations can be proportional and leave β non-identifiable even with exact derivatives; that is a correctness gap, not a circularity, and I do not count it toward the circularity score. The self-citations in the paper (Podina et al.) are for problem setup and empirical robustness claims, not for the uniqueness arguments, so they are not load-bearing. Overall, the central exact theorem has independent content, but the only quantitative error guarantee connecting it to the noisy, unevenly sampled experiments reduces to a self-referential interval computation, giving partial circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The theoretical results rest on strong data assumptions: exact derivatives, exact or near-exact repeated hidden states with distinct known multipliers, and known Lipschitz constants. No new physical entities are introduced. The error bounds additionally depend on Lipschitz constants and on a flawed interval computation, so the ledger burdens the practical claims more than the exact identifiability core.

free parameters (3)

Lipschitz constant L of u = not estimated
Theorem 3.5 and Remark 3.6 require L as an input; the paper does not show how a practitioner obtains it.
Lipschitz constant L1 of g = not estimated
Theorem 3.7 and Remark 3.8 require L1 as an input; no estimation procedure is given.
Lipschitz constant L2 of u = not estimated
Theorem 3.7 and Remark 3.8 require L2 as an input; no estimation procedure is given.

assumptions (5)

domain assumption The derivative ẋ of the trajectory is known exactly at each sampled state
Theorems 3.1-3.7 treat ẋ_i as data; Sections 4.1-4.3 explicitly feed true derivatives to the network, and the later autodiff approximation introduces unmodeled error.
domain assumption There exist two sample points with exactly equal y = H1(x) in the exact theorems, or D-close y in the approximate theorems, with C(xi) ≠ C(xj)
This is the core data condition; the paper acknowledges exact equality is unlikely and introduces D-close versions.
domain assumption The unknown functions u (and g in Theorem 3.7) are Lipschitz with known constants L (and L1, L2)
Needed for the Theorem 3.5 and 3.7 bounds; no estimation procedure is described.
domain assumption The known functions g, C, d, and H1 are known exactly
The theorems assume these are given, which is the typical setup but restricts the generality claimed by the title.
ad hoc to paper Neural network training converges to the global minimum of the MSE loss
The experiments claim recovery of the true solution; the theorems guarantee identifiability of the algebraic problem, not that gradient-based training reaches the identified solution.

how reviews work

0 comments

Cite this review

Pith. "Pith review of A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations." pith.science (2026). https://pith.science/paper/T6I56S5C

@misc{pith2026250516996,
  author       = {Pith},
  title        = {Pith review of: A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6I56S5C}},
  note         = {Machine review of arXiv:2505.16996}
}

read the original abstract

Inverse problems involving differential equations often require identifying unknown parameters or functions from data. Existing approaches, such as Physics-Informed Neural Networks (PINNs), Universal Differential Equations (UDEs) and Universal Physics-Informed Neural Networks (UPINNs), are effective at isolating either parameters or functions but can face challenges when applied simultaneously due to solution non-uniqueness. In this work, we introduce a framework that addresses these limitations by establishing conditions under which unique solutions can be guaranteed. To illustrate, we apply it to examples from biological systems and ecological dynamics, demonstrating accurate and interpretable results. Our approach significantly enhances the potential of machine learning techniques in modeling complex systems in science and engineering.

Figures

Figures reproduced from arXiv: 2505.16996 by the authors.

**Figure 3.** Solution to the Lotka-Volterra system with the given initial conditions (left). Comparison between the true and predicted values of βx and δy (right). The results demonstrate the effectiveness of the proposed methods in learning both unknown constants and functional forms. In the chemotherapy intervention problem, the network accurately recovered the true values of β and u(N), although the performance was sensitive… view at source ↗

**Figure 1.** Top row: Solutions to the differential equations and the comparison between true and predicted drug action for Trial 1 (u(N) = N). Bottom row: Results for Trial 2 (u(N) = N 2 ) [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** Results for the modified chemotherapy intervention problem. Left: Solution to the differential equations. Right: Comparison of the true and predicted growth term Ψ(N) and drug action u(N). 5.3. Lotka-Volterra Predator-Prey System The Lotka-Volterra system provided an additional test case for the uniqueness conditions in a more complex setting. The solutions of the system with the given initial conditions are shown… view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

17 extracted references · 6 canonical work pages

[1]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...
[2]

Inverse Physics-Informed Neural Networks for transport models in porous materials

Marco Berardi, Fabio Difonzo, and Matteo Icardi. Inverse Physics - Informed Neural Networks for transport models in porous materials, November 2024. URL http://arxiv.org/abs/2407.10654. arXiv:2407.10654 [math]

work page Pith review arXiv 2024
[3]

Brunton, Joshua L

Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113 0 (15): 0 3932--3937, April 2016. doi:10.1073/pnas.1517384113. URL https://www.pnas.org/doi/10.1073/pnas.1517384113. Publisher: Proceedings of the ...

doi:10.1073/pnas.1517384113 2016
[4]

Atefe Darabi, Zheming An, Muhammad Ali Al-Radhawi, William Cho, Milad Siami, and Eduardo D. Sontag. Combining model-based and data-driven models: an application to synthetic biology resource competition, March 2025. URL https://www.biorxiv.org/content/10.1101/2025.03.09.642275v1. Pages: 2025.03.09.642275 Section: New Results

work page doi:10.1101/2025.03.09.642275v1 2025
[5]

Kutz, and Steven Brunton

Brian de Silva, Kathleen Champion, Markus Quade, Jean-Christophe Loiseau, J. Kutz, and Steven Brunton. Pysindy: A python package for the sparse identification of nonlinear dynamical systems from data. Journal of Open Source Software, 5 0 (49): 0 2104, 2020. doi:10.21105/joss.02104. URL https://doi.org/10.21105/joss.02104

doi:10.21105/joss.02104 2020
[6]

Manuel Gloeckler, Michael Deistler, Christian Weilbach, Frank Wood, and Jakob H. Macke. All-in-one simulation-based inference, July 2024. URL http://arxiv.org/abs/2404.09636. arXiv:2404.09636 [cs]

arXiv 2024
[7]

Neural Operator : Learning Maps Between Function Spaces , May 2024

Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural Operator : Learning Maps Between Function Spaces , May 2024. URL http://arxiv.org/abs/2108.08481. arXiv:2108.08481 [cs]

arXiv 2024
[8]

Lingsch, Dana Grund, Siddhartha Mishra, and Georgios Kissas

Levi E. Lingsch, Dana Grund, Siddhartha Mishra, and Georgios Kissas. FUSE : Fast Unified Simulation and Estimation for PDEs . Advances in Neural Information Processing Systems, 37: 0 21656--21698, December 2024. URL https://proceedings.neurips.cc/paper_files/paper/2024/hash/266c0f191b04cbbbe529016d0edc847e-Abstract-Conference.html

work page 2024

Show all 17 references

[9]

Disentangled Representation Learning for Parametric Partial Differential Equations , October 2024

Ning Liu, Lu Zhang, Tian Gao, and Yue Yu. Disentangled Representation Learning for Parametric Partial Differential Equations , October 2024. URL http://arxiv.org/abs/2410.02136. arXiv:2410.02136 [cs]

2024
[10]

Universal Physics - Informed Neural Networks : Symbolic Differential Operator Discovery with Sparse Data

Lena Podina, Brydon Eastman, and Mohammad Kohandel. Universal Physics - Informed Neural Networks : Symbolic Differential Operator Discovery with Sparse Data . In Proceedings of the 40th International Conference on Machine Learning , pages 27948--27956. PMLR, July 2023. URL htt...

2023
[11]

Learning Chemotherapy Drug Action via Universal Physics - Informed Neural Networks , April 2024

Lena Podina, Ali Ghodsi, and Mohammad Kohandel. Learning Chemotherapy Drug Action via Universal Physics - Informed Neural Networks , April 2024. URL http://arxiv.org/abs/2404.08019. arXiv:2404.08019 [q-bio]

2024 arXiv
[12]

Universal Differential Equations for Scientific Machine Learning , November 2021

Christopher Rackauckas, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, Ali Ramadhan, and Alan Edelman. Universal Differential Equations for Scientific Machine Learning , November 2021. URL http://arxiv.org/abs/2001.04385. arXiv:2001.0...

2021 arXiv
[13]

Raissi, P

M. Raissi, P. Perdikaris, and G.E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378: 0 686--707, February 2019. ISSN 00219...

2019 doi
[14]

Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations

Maziar Raissi, Alireza Yazdani, and George Em Karniadakis. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Science, 367 0 (6481): 0 1026--1030, February 2020. ISSN 0036-8075, 1095-9203. doi:10.1126/science.aaw4741. URL https://www.scienc...

2020 doi
[15]

Physics- Informed Neural Networks and Extensions , August 2024

Maziar Raissi, Paris Perdikaris, Nazanin Ahmadi, and George Em Karniadakis. Physics- Informed Neural Networks and Extensions , August 2024. URL http://arxiv.org/abs/2408.16806. arXiv:2408.16806 [cs]

2024 arXiv
[16]

Rudy, Steven L

Samuel H. Rudy, Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Data-driven discovery of partial differential equations. Science Advances, 3 0 (4): 0 e1602614, April 2017. doi:10.1126/sciadv.1602614. URL https://www.science.org/doi/10.1126/sciadv.1602614. Publisher: ...

2017 doi
[17]

From PINNs to PIKANs : Recent Advances in Physics - Informed Machine Learning , October 2024

Juan Diego Toscano, Vivek Oommen, Alan John Varghese, Zongren Zou, Nazanin Ahmadi Daryakenari, Chenxi Wu, and George Em Karniadakis. From PINNs to PIKANs : Recent Advances in Physics - Informed Machine Learning , October 2024. URL http://arxiv.org/abs/2410.13228. arXiv:2410.13228 [cs]

2024 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...

[2] [2]

Inverse Physics-Informed Neural Networks for transport models in porous materials

Marco Berardi, Fabio Difonzo, and Matteo Icardi. Inverse Physics - Informed Neural Networks for transport models in porous materials, November 2024. URL http://arxiv.org/abs/2407.10654. arXiv:2407.10654 [math]

work page Pith review arXiv 2024

[3] [3]

Brunton, Joshua L

Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113 0 (15): 0 3932--3937, April 2016. doi:10.1073/pnas.1517384113. URL https://www.pnas.org/doi/10.1073/pnas.1517384113. Publisher: Proceedings of the ...

doi:10.1073/pnas.1517384113 2016

[4] [4]

Atefe Darabi, Zheming An, Muhammad Ali Al-Radhawi, William Cho, Milad Siami, and Eduardo D. Sontag. Combining model-based and data-driven models: an application to synthetic biology resource competition, March 2025. URL https://www.biorxiv.org/content/10.1101/2025.03.09.642275v1. Pages: 2025.03.09.642275 Section: New Results

work page doi:10.1101/2025.03.09.642275v1 2025

[5] [5]

Kutz, and Steven Brunton

Brian de Silva, Kathleen Champion, Markus Quade, Jean-Christophe Loiseau, J. Kutz, and Steven Brunton. Pysindy: A python package for the sparse identification of nonlinear dynamical systems from data. Journal of Open Source Software, 5 0 (49): 0 2104, 2020. doi:10.21105/joss.02104. URL https://doi.org/10.21105/joss.02104

doi:10.21105/joss.02104 2020

[6] [6]

Manuel Gloeckler, Michael Deistler, Christian Weilbach, Frank Wood, and Jakob H. Macke. All-in-one simulation-based inference, July 2024. URL http://arxiv.org/abs/2404.09636. arXiv:2404.09636 [cs]

arXiv 2024

[7] [7]

Neural Operator : Learning Maps Between Function Spaces , May 2024

Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural Operator : Learning Maps Between Function Spaces , May 2024. URL http://arxiv.org/abs/2108.08481. arXiv:2108.08481 [cs]

arXiv 2024

[8] [8]

Lingsch, Dana Grund, Siddhartha Mishra, and Georgios Kissas

Levi E. Lingsch, Dana Grund, Siddhartha Mishra, and Georgios Kissas. FUSE : Fast Unified Simulation and Estimation for PDEs . Advances in Neural Information Processing Systems, 37: 0 21656--21698, December 2024. URL https://proceedings.neurips.cc/paper_files/paper/2024/hash/266c0f191b04cbbbe529016d0edc847e-Abstract-Conference.html

work page 2024