prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What did Justice Kennedy argue about Quill in Direct Marketing Ass'n v. Brohl?

Read the following text and answer briefly.

Context 1: South Dakota v  Wayfair, Inc  - Harvard Law Review
Fourth Circuit Invalidates Maryland Statute Regulating Price Gouging in the Sale of Generic Drugs 
South Dakota Supreme Court Holds Unconstitutional State Law Requiring Internet Retailers Without In-State Physical Presence to Remit Sales Tax 
Judicial junk, the Court has long thought, is easier to scrap when the erroneous precedent cannot be fixed by Congress, as in constitutional cases 1× 1  See Burnet v  Coronado Oil & Gas Co , 285 U S  393, 405–10 (1932) (Brandeis, J , dissenting); Lee Epstein, William M  Landes & Adam Liptak, The Decision to Depart (or Not) from Constitutional Precedent: An Empirical Study of the Roberts Court, 90 N Y U  L  Rev  1115, 1116 (2015) (“[Justice Brandeis’s] dissenting opinion       now has the status of black letter law ”)  On the flip side, whenever a bad precedent can be corrected by Congress, stare decisis applies with “special force ”2× 2  See Patterson v  McLean Credit Union, 491 U S  164, 172–73 (1989)  The Court, following Justice Brandeis, usually articulates the rule as distinguishing between “constitutional” and “statutory” precedents  See, e g , id  But the distinction is occasionally said to be between “constitutional” and “nonconstitutional cases ” See, e g , Glidden Co  v  Zdanok, 370 U S  530, 543 (1962) (plurality opinion)  Nomenclature aside, the Court has — until now — adhered to Justice Brandeis’s key insight that the important factor is whether or not the mistake may be legislatively corrected  Last Term, in South Dakota v  Wayfair, Inc ,3× 3  138 S  Ct  2080 (2018)  the Court tinkered with this thinking in overruling an outdated dormant commerce clause precedent  Dormant commerce clause decisions technically produce constitutional holdings, but Congress may override them at will 4× 4  See Prudential Ins  Co  v  Benjamin, 328 U S  408, 421–27 (1946)  Under the usual logic of stare decisis, it should take special force to dislodge such precedents  But Wayfair applied the weakened stare decisis of constitutional cases, asserting that the Court must “address a false constitutional premise         whether or not Congress can or will act ”5× 5  Wayfair, 138 S  Ct  at 2096–97 
Emerging from Wayfair is an odd and ominous development in stare decisis doctrine  Odd, because it turns on a formal classification instead of on Congress’s practical ability to fix the problem  Ominous, because the Court’s logic leads far past the dormant commerce clause  Wayfair grants only feeble stare decisis to precedents that set a “constitutional default rule,”6× 6  Id  at 2096 (“While       Congress has the authority to change the physical presence rule, Congress cannot change the constitutional default rule ”)  meaning constitutional decisions that allow for legislative adjustment or override  This new stare decisis analysis makes other precedents setting constitutional default rules more vulnerable — including, perhaps, mainstays of criminal procedure like Miranda v  Arizona7× 7  384 U S  436 (1966)  and Mapp v  Ohio 8× 8  367 U S  643 (1961) 
Since its 1967 decision in National Bellas Hess, Inc  v  Department of Revenue,9× 9  386 U S  753 (1967)  the Court has held that, under the “dormant” or “negative” implication of the Commerce Clause,10× 10  The dormant or negative commerce clause is a judicial derivation from the Commerce Clause “prohibiting States from discriminating against or imposing excessive burdens on interstate commerce without congressional approval,” which “strikes at one of the chief evils that led to the adoption of the Constitution, namely, state tariffs and other laws that burdened interstate commerce ” Comptroller of the Treasury of Md  v  Wynne, 135 S  Ct  1787, 1794 (2015)  states may not compel remote sellers with no physical presence in the state to collect and remit sales taxes 11× 11  See Bellas Hess, 386 U S  at 759–60  In Quill Corp  v  North Dakota,12× 12  504 U S  298 (1992)  the Court refused to overrule the “bright-line, physical-presence requirement” of Bellas Hess, leaning heavily on stare decisis 13× 13  Id  at 317–18  Three Justices joined a concurrence explaining that their decision rested solely “on the basis of stare decisis ” Id  at 320 (Scalia, J , concurring in part and concurring in the judgment)  So the physical presence test remained the law of the land while the internet conquered the earth  Justice Kennedy had joined the Quill majority and Justice Scalia’s concurring opinion emphasizing stare decisis, but by 2015 he had second thoughts  Writing separately in Direct Marketing Ass’n v  Brohl,14× 14  135 S  Ct  1124 (2015)  Justice Kennedy acknowledged that “[t]he Internet has caused far-reaching systemic and structural changes in the economy” and therefore “Quill now harms States to a degree far greater than could have been anticipated earlier ”15× 15  Id  at 1135 (Kennedy, J , concurring)  He concluded with the wish that “[t]he legal system should find an appropriate case for this Court to reexamine Quill and Bellas Hess ”16× 16  Id 
Seldom has a concurring opinion signed by a lone Justice prompted a state to officially declare an emergency  Yet in 2016, in response to Justice Kennedy’s overture, the South Dakota legislature passed a law, S B  106, “to provide for the collection of sales taxes from certain remote sellers       and to declare an emergency ”17× 17  2016 S D  Sess  Laws ch  70 pmbl  217 (codified at S D  Codified Laws § 10-64 (2017))  It required every remote seller to collect and remit sales tax if the seller’s business in South Dakota comprised either a “gross revenue” greater than $100,000 or at least 200 “separate transactions” within one calendar year 18× 18  Id  § 1  Significantly, the law did not apply retroactively 19× 19  Id  § 5  The “emergency” declaration was necessary to give the law immediate effect, for the purpose of “permitting the most expeditious possible review of the constitutionality of this law” by the U S  Supreme Court 20× 20  Id  § 8(8)  As Justice Alito put it, the “South Dakota law [was] obviously a test case ”21× 21  Transcript of Oral Argument at 27, Wayfair, 138 S  Ct  2080 (No  17-494), https://www supremecourt gov/oral_arguments/argument_transcripts/2017/17-494_7lho pdf [https://perma cc/8HYH-VU8N] 
Expeditiously, a group of remote sellers challenged the law  After being sued by South Dakota for refusing to register for the newly required sales tax license, Wayfair, Inc , Overstock com, Inc , and Newegg, Inc  moved for summary judgment in South Dakota circuit court on the grounds that S B  106 was unconstitutional under Quill and Bellas Hess — a point South Dakota conceded, indicating that it was seeking review by the U S  Supreme Court to overturn Quill 22× 22  State v  Wayfair Inc , 2017 SD 56, ¶¶ 9–11, 901 N W 2d 754, 759–60  Accordingly, the South Dakota circuit court granted the motion for summary judgment and South Dakota appealed to the state’s highest court 23× 23  Id  ¶ 12, 901 N W 2d at 760  The South Dakota Supreme Court unanimously affirmed, recognizing that South Dakota’s “arguments on the merits” may be “persuasive” but “Quill remains the controlling precedent ”24× 24  Id  ¶ 18, 901 N W 2d at 761  See generally Recent Case, State v  Wayfair Inc , 2017 SD 56, 901 N W 2d 754 (S D  2017), 131 Harv  L  Rev  2089 (2018) 
The U S  Supreme Court vacated and remanded 25× 25  Wayfair, 138 S  Ct  at 2100  Writing for the Court one last time, Justice Kennedy26× 26  Justices Thomas, Ginsburg, Alito, and Gorsuch joined Justice Kennedy’s opinion  pilloried Quill’s physical presence rule as “arbitrary, formalistic,” “anachronistic,” and “unfair and unjust” to both states and brick-and-mortar retailers 27× 27  Wayfair, 138 S  Ct  at 2092, 2095  After all, the rationale of Quill was that remote sellers lacked a sufficiently “substantial nexus” with the state to justify imposing a duty of tax collection 28× 28  Quill Corp  v  North Dakota, 504 U S  298, 311 (1992) (quoting Complete Auto Transit, Inc  v  Brady, 430 U S  274, 279 (1977))  This was wrong even in the mail-order catalog days of 1967 and 1992, but “the Internet revolution has made [Quill’s] earlier error all the more egregious and harmful ”29× 29  Wayfair, 138 S  Ct  at 2097; see also id  at 2092  The rule deprived the states of billions of dollars, since they could not force remote sellers to collect the tax and consumers hardly ever paid it on their own 30× 30  Id  at 2088 (“[C]onsumer compliance rates are notoriously low ”)  Quill “serve[d] as a judicially created tax shelter” for remote retailers who do a great deal of business online 31× 31  Id  at 2094 
Satisfied that Bellas Hess and Quill were wrongly decided, the Court then jumped the hurdle of stare decisis  The Quill Court had feared upsetting reliance interests 32× 32  Quill, 504 U S  at 317 (“Bellas Hess       has engendered substantial reliance and has become part of the basic framework of a sizable industry ”)  Wayfair shrugged off this concern, noting that “stare decisis accommodates only ‘legitimate reliance interest[s]’”; by contrast, reliance on the physical presence rule was largely due to consumers evading their use-tax obligations 33× 33  Wayfair, 138 S  Ct  at 2098 (alteration in original) (quoting United States v  Ross, 456 U S  798, 824 (1982))  Quill had also appealed to Congress’s ultimate authority over interstate commerce as a reason to abide by a precedent, even if wrongly decided 34× 34  See Quill, 504 U S  at 318–19; id  at 320 (Scalia, J , concurring in part and concurring in the judgment) (“Congress       can change the rule of Bellas Hess by simply saying so ”)  But Wayfair denied that Congress’s ability to change the law was a proper consideration:
While it can be conceded that Congress has the authority to change the physical presence rule, Congress cannot change the constitutional default rule  It is inconsistent with the Court’s proper role to ask Congress to address a false constitutional premise of this Court’s own creation  Courts have acted as the front line of review in this limited sphere; and hence it is important that their principles be accurate and logical, whether or not Congress can or will act in response 35× 35  Wayfair, 138 S  Ct  at 2096–97 
Having dispensed with the physical presence rule, the Court remanded the case to the South Dakota courts to determine in the first instance “whether some other principle in the Court’s Commerce Clause doctrine might invalidate the Act ”36× 36  Id  at 2099  But the Court listed “several features [of South Dakota law] that appear[ed] designed to prevent discrimination against or undue burdens upon interstate commerce ” Id 
Justices Thomas and Gorsuch each filed concurring opinions  Justice Thomas wistfully likened himself to Justice White — who voted for Bellas Hess but against Quill a quarter-century later — and confessed that he “should have joined [Justice White’s dissenting] opinion ”37× 37  Id  at 2100 (Thomas, J , concurring)  Justice Thomas added that the “Court’s entire negative Commerce Clause jurisprudence” is wrong and should be abandoned 38× 38  Id  Justice Gorsuch also wrote separately to express skepticism of the Court’s dormant commerce clause jurisprudence, raising “questions for another day” of whether the doctrine “can be squared with the text of the Commerce Clause, justified by stare decisis, or defended as misbranded products of federalism or antidiscrimination imperatives flowing from Article IV’s Privileges and Immunities Clause ”39× 39  Id  at 2100–01 (Gorsuch, J , concurring) 
Chief Justice Roberts dissented 40× 40  Justices Breyer, Sotomayor, and Kagan joined the Chief Justice’s dissent  Surprisingly, the dissenting Justices “agree[d] that Bellas Hess was wrongly decided, for many of the reasons given by the Court ”41× 41  Wayfair, 138 S  Ct  at 2101 (Roberts, C J , dissenting)  The dispute between the majority and the dissent turned entirely on the principles and application of stare decisis  Chief Justice Roberts argued that whether or how to reverse Quill should be left to Congress, which “has the flexibility to address these questions in a wide variety of ways” and “can focus directly on current policy concerns rather than past legal mistakes ”42× 42  Id  at 2104  He also pointed to the “baffling” burdens of compliance with the idiosyncratic tax codes of “[o]ver 10,000 jurisdictions,” particularly for small businesses, and doubted that new “software” — the majority’s proposed solution to this mess43× 43  Id  at 2098 (majority opinion) (“Eventually, software that is available at a reasonable cost may make it easier for small businesses to cope with these problems ”)  — would soon solve the problem 44× 44  Id  at 2103–04 (Roberts, C J , dissenting)  In Bellas Hess, the Court reasoned that the dormant commerce clause protects interstate business from being “entangle[d]       in a virtual welter of complicated obligations to local jurisdictions ” Nat’l Bellas Hess, Inc  v  Dep’t of Revenue, 386 U S  753, 759–60 (1967)  The dissent replied that the Court “vastly underestimate[d] the skill of contemporary man and his machines ” Id  at 766 (Fortas, J , dissenting)  The dispute in Wayfair over whether software is up to the task effectively reprised the old debate from Bellas Hess, only this time couched as part of the stare decisis inquiry’s concern for reliance interests rather than as a matter of dormant commerce clause doctrine  While Wayfair acknowledged that “[c]omplex state tax systems could have the effect of discriminating against interstate commerce,” 138 S  Ct  at 2099, the Court remarked that “[t]he physical presence rule is a poor proxy” for an inquiry into any actual burdens imposed on interstate commerce, id  at 2093 
Chief Justice Roberts emphasized that a “heightened form of stare decisis”45× 45  Wayfair, 138 S  Ct  at 2102 (Roberts, C J , dissenting)  applies when “Congress       can, if it wishes, override this Court’s decisions with contrary legislation ”46× 46  Id  at 2101 (first citing Michigan v  Bay Mills Indian Cmty , 134 S  Ct  2024, 2036 (2014) (tribal sovereign immunity); then citing Kimble v  Marvel Entm’t, LLC, 135 S  Ct  2401, 2409 (2015) (statutory interpretation); and then citing Halliburton Co  v  Erica P  John Fund, Inc , 134 S  Ct  2398, 2411 (2014) (judicially created doctrine implementing a judicially created cause of action))  In Quill, the Chief Justice noted, the Court had taken to heart that “Congress may be better qualified” and “has the ultimate power to resolve” the question47× 47  Id  at 2102 (quoting Quill Corp  v  North Dakota, 504 U S  279, 318 (1992))  while Justice Scalia had “recogniz[ed] that stare decisis has ‘special force’ in the dormant Commerce Clause context due to Congress’s ‘final say over regulation of interstate commerce ’”48× 48  Id  (quoting Quill, 504 U S  at 320 (Scalia, J , concurring in part and concurring in the judgment))  Moreover, “[i]f stare decisis applied with special force in Quill, it should be an even greater impediment” afterward since Quill effectively “tossed [the ball] into Congress’s court ”49× 49  Id  (alteration in original) (quoting Kimble, 135 S  Ct  at 2409); cf  Bay Mills, 134 S  Ct  at 2039 n 12 (“When we inform Congress that it has primary responsibility over a sphere of law, and invite Congress to consider a specific issue within that sphere, we cannot deem irrelevant how Congress responds ”)  Because the Court invited Congress to act and then “suddenly chang[ed] the ground rules, the Court may have waylaid Congress’s consideration of the issue ”50× 50  Wayfair, 138 S  Ct  at 2102–03 (Roberts, C J , dissenting) 
In Wayfair, the Court applied the flimsier form of stare decisis to a precedent that could have been overruled by Congress  It did so in the context of a dormant commerce clause case, but Wayfair’s logic extends to all constitutional default rules — that is, constitutional decisions that Congress remains free to change  Not only does Wayfair deviate from the Court’s decades-old stare decisis analysis, it also imperils other precedents that set constitutional default rules 
The Court’s reasoning in Wayfair departs from its prior stare decisis analysis  In 1932, Justice Brandeis posited that stare decisis must bend “in cases involving the Federal Constitution, where correction through legislative action is practically impossible ”51× 51  Burnet v  Coronado Oil & Gas Co , 285 U S  393, 406–07 (1932) (Brandeis, J , dissenting)  The Court has long since adopted his argument,52× 52  See, e g , Smith v  Allwright, 321 U S  649, 665 (1944)  as well as its corollary — that stare decisis commands “special force in the area of statutory interpretation” where “Congress remains free to alter what [the Court has] done ”53× 53  Patterson v  McLean Credit Union, 491 U S  164, 172–73 (1989)  For normative evaluations of heightened stare decisis for statutory precedents, see generally Einer Elhauge, Statutory Default Rules: How to Interpret Unclear Legislation 211–23 (2008); and William N  Eskridge, Jr , Overruling Statutory Precedents, 76 Geo  L J  1361, 1364–1409 (1988)  Justice Brandeis’s logic demands that dormant commerce clause cases, where Congress is free to act, be granted the weightier stare decisis 54× 54  Scholars have noted the curious fact that Justice Brandeis included many dormant commerce clause cases as examples of overruled constitutional precedents  See, e g , Earl M  Maltz, Commentary, Some Thoughts on the Death of Stare Decisis in Constitutional Law, 1980 Wis  L  Rev  467, 468–469, 469 n 11  One explanation for this is that Justice Brandeis sought the authority of Chief Justice Taney’s dictum that the Court’s “opinion upon the construction of the Constitution is always open to discussion” — which referred to the dormant commerce clause  See Burnet, 285 U S  at 408 n 3 (Brandeis, J , dissenting) (quoting The Passenger Cases, 48 U S  (7 How ) 283, 470 (1849) (Taney, C J , dissenting))  In Chief Justice Taney’s time, it was thought that Congress could not override the Court’s dormant commerce clause decisions, see Cooley v  Bd  of Wardens, 53 U S  (12 How ) 299, 321 (1852), so the context of Chief Justice Taney’s dictum does not conflict with Justice Brandeis’s theory of stare decisis  The Court applied this reasoning in Quill, as Chief Justice Roberts underscored 55× 55  Wayfair, 138 S  Ct  at 2102 (Roberts, C J , dissenting) 
Yet the Wayfair majority refused to consider Congress’s authority to legislate as a relevant factor for stare decisis 56× 56  Even Justice Kennedy’s earlier opinion in Direct Marketing contemplated judicially overruling Quill, conspicuously neglecting a possible legislative solution  See supra p  278  The Court even insisted that to do so “is inconsistent with the Court’s proper role,” since Quill embodied “a false constitutional premise of th[e] Court’s own creation ”57× 57  Wayfair, 138 S  Ct  at 2096 (emphasis added)  This refusal breaks from the practical Brandeisian wisdom that has guided the Court’s treatment of precedent for the better part of a century  The point is not that stare decisis should have ultimately propped up Bellas Hess yet again, as Wayfair’s dissenting Justices maintained  After all, a realistic approach that is alert to each branch’s institutional capacities might have led to the conclusion that Congress was actually ill-equipped to overrule Quill  In this vein, the Court could have sensibly pointed out that Congress is unlikely to stick its neck out with a tax hike (or a look-alike) from which only the states would benefit 58× 58  For two practical arguments to this effect, see Brian Galle, Essay, Kill Quill, Keep the Dormant Commerce Clause: History’s Lessons on Congressional Control of State Taxation, 70 Stan  L  Rev  Online 158, 160–62 (2018), https://review law stanford edu/wp-content/uploads/sites/3/2018/03/70-Stan -L -Rev -Online-158-Galle pdf [https://perma cc/22YP-P4V5]; Edward A  Zelinsky, The Political Process Argument for Overruling Quill, 82 Brook  L  Rev  1177, 1191–92 (2017)  Indeed, South Dakota advanced such practical arguments in its brief 59× 59  See Petitioner’s Brief at 54, Wayfair, 138 S  Ct  2080 (No  17-494) (“Congress has little incentive to act here because it would be (or appear to be) authorizing new or greater tax collections from its constituents, while receiving none of the revenue in return ”)  More generally, the Court might have discussed the limits of the states’ influence in the federal system as a reason not to wait for congressional intervention, a topic it has debated on other occasions 60× 60  See Richard H  Pildes, Institutional Formalism and Realism in Constitutional and Public Law, 2013 Sup  Ct  Rev  1, 30–32; see also Galle, supra note 58, at 159 (“Congress is not a trustworthy guardian of state fiscal power, making continuing judicial involvement a more appealing prospect ”)  Or it could have argued that new facts on the ground — namely, the blast of e-commerce that hit like a comet after Quill — overpowered stare decisis of any force, special or plain 61× 61  Two recent studies of stare decisis highlighted the physical presence rule as exemplifying a precedent that may reasonably be overruled due to changed facts  See Bryan A  Garner et al , The Law of Judicial Precedent 364–65 (2016); Randy J  Kozel, Settled Versus Right: A Theory of Precedent 112–13 (2017)  It should be noted that the authors of The Law of Judicial Precedent classify the physical presence rule as a constitutional precedent for stare decisis purposes, thus anticipating the Court’s misstep in Wayfair  Garner et al , supra, at 354–65  Because even statutory precedents may sometimes be overruled,62× 62  See Patterson v  McLean Credit Union, 491 U S  164, 173–74 (1989) (discussing justifications for overruling statutory precedents)  Contra Lawrence C  Marshall, “Let Congress Do It”: The Case for an Absolute Rule of Statutory Stare Decisis, 88 Mich  L  Rev  177 (1989)  the Court could have killed Quill without first planting its constitutional kiss of death 63× 63  Cf  Thomas R  Lee, Stare Decisis in Historical Perspective: From the Founding Era to the Rehnquist Court, 52 Vand  L  Rev  647, 704 (1999) (“Justice Brandeis’       memorable prose has since become a mandatory part of the burial rite for any constitutional precedent ”) 
The Court resisted such arguments  Instead, Wayfair reasoned that Congress’s total ability to correct an erroneous decision counts for nothing when the Court gets the Constitution wrong  That such a theory sprouts from a case like Wayfair, which repudiated a “formalistic distinction,”64× 64  Wayfair, 138 S  Ct  at 2092  is ironic  Wayfair’s stare decisis analysis resorts to the formalism of making constitutional a “magic” word65× 65  See Transcript of Oral Argument, supra note 21, at 12  rather than asking whether Congress can step in 
Moreover, the Court’s new thinking on stare decisis threatens other constitutional default rules  Wayfair now stands for the proposition that a “constitutional default rule” — a term the Court apparently lifted from South Dakota’s reply brief on the merits66× 66  Reply Brief at 22, Wayfair, 138 S  Ct  2080 (No  17-494) (“Congress is polarized, which makes it critical       to get the constitutional default rule right ”)  — gets only weakened stare decisis  To appreciate why this holding matters, it is worth exploring the concept and scope of constitutional default rules  Contract theory describes default rules as legal rules that the parties may “contract around ”67× 67  See, e g , Ian Ayres & Robert Gertner, Filling Gaps in Incomplete Contracts: An Economic Theory of Default Rules, 99 Yale L J  87, 87 (1989)  Although “constitutional default rule” could be read broadly to include a variety of actors and contracting mechanisms,68× 68  See John Ferejohn & Barry Friedman, Toward a Political Theory of Constitutional Default Rules, 33 Fla  St  U  L  Rev  825, 826 (2006) (“When we speak of default rules in constitutional law, we typically are talking about specifications of ways the government can act (or modify its behavior) to get around a constitutional prohibition ”)  the Court’s use of the term for purposes of stare decisis may be narrowly defined as judicial precedents of constitutional law that “are ultimately subject to congressional control ”69× 69  Gillian E  Metzger, Congress, Article IV, and Interstate Relations, 120 Harv  L  Rev  1468, 1525 (2007) (describing judicially enforceable “constitutional default rules imposing obligations on the states in the name of union [that] are ultimately subject to congressional control”)  The dormant commerce clause is a paradigmatic constitutional default rule because what the Court does today Congress may undo tomorrow  Justice Scalia declared this fact “[t]he clearest sign that the negative Commerce Clause is a judicial fraud,” for “[h]ow could congressional consent lift a constitutional prohibition?”70× 70  Comptroller of the Treasury of Md  v  Wynne, 135 S  Ct  1787, 1808 (2015) (Scalia, J , dissenting)  But that’s what a constitutional default rule is  The Court has allowed Congress to overturn its dormant commerce clause cases since 1891 71× 71  See In re Rahrer, 140 U S  545, 560–62 (1891) 
Dormant commerce clause cases are not the only constitutional default rules  Professor Laurence Tribe’s treatise identifies two others 72× 72  1 Laurence H  Tribe, American Constitutional Law § 6-35 (3d ed  2000)  And in a groundbreaking article, Professor Henry Monaghan revealed “a substructure of substantive, procedural, and remedial rules” forming “a constitutional common law subject to amendment, modification, or even reversal by Congress ”73× 73  Henry P  Monaghan, The Supreme Court, 1974 Term — Foreword: Constitutional Common Law, 89 Harv  L  Rev  1, 2–3 (1975); see also Susan R  Klein, Identifying and (Re)Formulating Prophylactic Rules, Safe Harbors, and Incidental Rights in Constitutional Criminal Procedure, 99 Mich  L  Rev  1030 (2001) (further developing Monaghan’s theory in criminal procedure context)  What follows is a list of six lines of cases beyond the dormant commerce clause that may be fairly described as constitutional default rules  The first two are drawn from Tribe’s treatise while the next four are found in Monaghan’s article:
(1) State Taxation of Federal Instrumentalities: States may not tax instrumentalities of the federal government74× 74  McCulloch v  Maryland, 17 U S  (4 Wheat ) 316, 436 (1819)  — unless Congress consents 75× 75  See, e g , Helvering v  Gerhardt, 304 U S  405, 411 n 1 (1938) (“Congress may curtail an immunity which might otherwise be implied or enlarge it beyond the point where, Congress being silent, the Court would set its limits ” (citations omitted))  One court has described such judicial decisions as setting a “constitutional default rule ” United States v  Delaware, 958 F 2d 555, 560 n 9 (3d Cir  1992) (“[W]e must decide the constitutional default rule for this type of tax, fully aware that Congress could decide at any time to reverse our decision statutorily ”)  (2) Article I, Section 10 Cases: Article I, Section 10 provides that certain prohibitions on the states may be waived by Congress 76× 76  See U S  Const  art  I, § 10, cls  2–3  The Court has taken note of this when considering whether to overrule, for instance, an Import-Export Clause precedent 77× 77  See Hooven & Allison Co  v  Evatt, 324 U S  652, 668 (1945) (“In view of the fact that the Constitution gives Congress authority to consent to state taxation of imports and hence to lay down its own test for determining when the immunity ends, we see no convincing practical reason for abandoning the test which has been applied for more than a century        ”), overruled on other grounds by Limbach v  Hooven & Allison Co , 466 U S  353 (1984)  In Michelin Tire Corp  v  Wages, 423 U S  276 (1976), the Court left open the question whether “Congress may authorize, under the Import-Export Clause, an exaction that it could not directly impose under the Tax Clause ” Id  at 301 n 13  Metzger, however, argues that the Import-Export Clause is free of other clauses’ limits on congressional power  See Metzger, supra note 69, at 1500 & n 120  (3) Bivens Cases: In Bivens v  Six Unknown Named Agents of Federal Bureau of Narcotics,78× 78  403 U S  388 (1971)  the Court held that a violation of the Fourth Amendment gives rise to a right to sue for damages 79× 79  Id  at 397  But the Court has also held that “[s]uch a cause of action may be defeated       when       Congress has provided an alternative remedy which it explicitly declared to be a substitute for recovery directly under the Constitution and viewed as equally effective ”80× 80  Carlson v  Green, 446 U S  14, 18–19 (1980)  (4) Miranda Cases: The Miranda Court famously “encourage[d]” Congress and the states to explore alternative “procedures which are at least as effective in apprising accused persons of their right of silence and in assuring a continuous opportunity to exercise it ”81× 81  Miranda v  Arizona, 384 U S  436, 467 (1966)  In Dickerson v  United States, 530 U S  428 (2000), the Court struck down a congressional attempt to effectively abolish Miranda, holding that “Miranda announced a constitutional rule that Congress may not supersede legislatively ” Id  at 444  But Dickerson also stood by Miranda’s “invitation for legislative action” to replace Miranda with an adequate substitute  Id  at 440; see also Michael C  Dorf & Barry Friedman, Shared Constitutional Interpretation, 2000 Sup  Ct  Rev  61 (discussing legislative alternatives to Miranda)  (5) The Police Lineup Case: In United States v  Wade,82× 82  388 U S  218 (1967)  the Court created an exclusionary rule for evidence obtained from a police lineup in violation of the Sixth Amendment right to counsel but acknowledged that it could be replaced by “[l]egislative or other regulations       which eliminate the risks of abuse ”83× 83  Id  at 239  (6) The Exclusionary Rule Cases: Mapp v  Ohio made the Fourth Amendment “exclusionary rule” binding on the states,84× 84  367 U S  643, 655 (1961)  yet Congress is thought to have the power to replace it 85× 85  See Bivens v  Six Unknown Named Agents of Fed  Bureau of Narcotics, 403 U S  388, 422–24 (1971) (Burger, C J , dissenting) (inviting Congress to replace the Fourth Amendment exclusionary rule); Harold J  Krent, How to Move Beyond the Exclusionary Rule: Structuring Judicial Response to Legislative Reform Efforts, 26 Pepp  L  Rev  855, 864–71 (1999) 
All of the above are arguably constitutional default rules set by the Court that remain, to one degree or another, open to congressional revision  The list could be longer or shorter, depending on which default rules the Court will view as constitutional86× 86  A shorter list could be produced by whittling away at the constitutional status of the cases identified by Monaghan  While the Court has held that Miranda is a constitutional decision, Dickerson, 530 U S  at 444, some of the other cases may be viewed as nonconstitutional  See, e g , Collins v  Virginia, 138 S  Ct  1663, 1675–80 (2018) (Thomas, J , concurring) (arguing that Mapp is “nonconstitutional,” id  at 1678 n 5); Richard H  Fallon, Jr  et al , Hart and Wechsler’s The Federal Courts and the Federal System 775–77 (7th ed  2015) (discussing whether Bivens is constitutionally required)  Conversely, a longer list might include any constitutional right that can be waived by a party  See, e g , Daniel A  Farber, Another View of the Quagmire: Unconstitutional Conditions and Contract Theory, 33 Fla  St  U  L  Rev  913, 918 (2006) (describing the Eleventh Amendment as “just a contractual default rule that the states are free to barter away”)  Such a list might also include various constitutionally inspired judicial presumptions  See, e g , Jack Goldsmith & John F  Manning, The President’s Completion Power, 115 Yale L J  2280, 2299 (2006) (describing the Chevron presumption of delegated interpretive power to administrative agencies as “a constitutionally inspired default rule”); Nicholas Quinn Rosenkranz, Federal Rules of Statutory Interpretation, 115 Harv  L  Rev  2085, 2097–98 (2002) (describing clear statement rules as “constitutional default rules” reversible by Congress)  Many other decisions could likely be characterized as constitutional default rules; the list above is only an initial stab  and on how it will answer open questions about congressional authority over certain constitutional provisions 87× 87  See, e g , Thomas v  Wash  Gas Light Co , 448 U S  261, 272 n 18 (1980) (plurality opinion) (leaving unresolved whether Congress may limit constitutional full faith and credit obligations); White v  Mass  Council of Constr  Emp’rs, Inc , 460 U S  204, 215 n 1 (1983) (Blackmun, J , concurring in part and dissenting in part) (leaving unresolved “whether Congress may authorize       what otherwise would be a violation” of the Privileges and Immunities Clause); 1 Tribe, supra note 72, § 6-35, at 1243–44 (arguing that Congress cannot override judicial constructions of the Privileges and Immunities Clause); Metzger, supra note 69, at 1486–89 (arguing the opposite)  But the takeaway is clear: weaker stare decisis for constitutional default rules  Pre-Wayfair, one would have thought that stare decisis applies with special force to such precedents, given congressional power to set them straight  Not anymore  Why? Because it is improper to “ask Congress to address a false constitutional premise of th[e] Court’s own creation ”88× 88  Wayfair, 138 S  Ct  at 2096  The Latin for Wayfair’s doctrine is not stare decisis, which should reflect a realistic, working relationship between the legislative and judicial branches  It is mea culpa 
In its zeal to update the Constitution for “the Cyber Age,”89× 89  Id  at 2097  the Court deleted Congress from stare decisis doctrine in constitutional cases  The Court had better options  It could have left Quill on Congress’s doorstep, as the dissent argued  Or it could have justified overruling Quill notwithstanding the special force of stare decisis  Instead, the Court reasoned that it doesn’t matter whether Congress is willing and able to do the job: a constitutional mess calls for a judicial clean-up crew  For constitutional default rules — a category of decisions embracing the dormant commerce clause and sweeping far beyond — Wayfair’s new theory of stare decisis makes the Court’s precedents less sticky and Congress less relevant 
Context 2: Paper Info

Title: Bistability between π-diradical open-shell and closed-shell states in indeno[1,2-a]fluorene
Publish Date: Unkown
Author List: Shantanu Mishra (from IBM Research Europe -Zurich), Manuel Vilas-Varela (from Department of Organic Chemistry, Center for Research in Biological Chemistry and Molecular Materials (CiQUS), University of Santiago de Compostela), Leonard-Alexander Lieske (from IBM Research Europe -Zurich), Ricardo Ortiz (from Donostia International Physics Center (DIPC)), Igor Rončević (from Department of Chemistry, University of Oxford), Florian Albrecht (from IBM Research Europe -Zurich), Diego Peña (from Department of Organic Chemistry, Center for Research in Biological Chemistry and Molecular Materials (CiQUS), University of Santiago de Compostela), Leo Gross (from IBM Research Europe -Zurich)

Figure

Fig  1 | Non-benzenoid non-alternant polycyclic conjugated hydrocarbons a, Classical nonbenzenoid non-alternant polycyclic conjugated hydrocarbons: pentalene, azulene and heptalene b, Generation of indacenes and indenoindenes through benzinterposition and benzannelation of pentalene, respectively Gray filled rings represent Clar sextets c, Closed-shell Kekulé (left) and openshell non-Kekulé (right) resonance structures of QDMs Note that meta-QDM is a non-Kekulé molecule All indenofluorene isomers, being derived through benzannelation of indacenes, contain a central QDM moiety d, Closed-shell Kekulé (top) and open-shell non-Kekulé (bottom) resonance structures of indenofluorenes Compared to their closed-shell structures, 1 and 5 gain two Clar sextets in the openshell structure, while 2-4 gain only one Clar sextet in the open-shell structure Colored bonds in d highlight the ortho-and para-QDM moieties in the two closed-shell Kekulé structures of 5  e, Scheme of on-surface generation of 5 by voltage pulse-induced dehydrogenation of 6 (C20H14) Structures 7 and 8 represent the two monoradical species (C20H13) 
Fig  2 | Characterization of open-shell indeno[1,2-a]fluorene on bilayer NaCl/Au(111) a, DFTcalculated wave functions of the frontier orbitals of 5OS in the triplet configuration for the spin up (occupied) level (isovalue: 0 002 e -Å -3 ) Blue and red colors represent opposite phases of the wave function b, Corresponding DFT-calculated spin density of 5OS (isovalue: 0 01 e -Å -3) Blue and orange colors represent spin up and spin down densities, respectively c, Probability density of the SOMOs of 5OS (isovalue: 0 001 e -Å -3 ) d, DFT-calculated bond lengths of 5OS e, Constant-height I(V) spectra acquired on a species of 5 assigned as 5OS, along with the corresponding dI/dV(V) spectra Open feedback parameters: V = -2 V, I = 0 17 pA (negative bias side) and V = 2 V, I = 0 17 pA (positive bias side) Acquisition position of the spectra is shown in Supplementary Fig 7 f, Scheme of many-body transitions associated to the measured ionic resonances of 5OS Also shown are STM images of assigned 5OS at biases where the corresponding transitions become accessible Scanning parameters: I = 0 3 pA (V = -1 2V and -1 5 V) and 0 2 pA (V = 1 3 V and 1 6 V)  g, Laplace-filtered AFM image of assigned 5OS STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3Å The tip-height offset Δz for each panel is provided with respect to the STM setpoint, and positive (negative) values of Δz denote tip approach (retraction) from the STM setpoint f and g show the same molecule at the same adsorption site, which is next to a trilayer NaCl island The bright and dark features in the trilayer NaCl island in g correspond to Cl -and Na + ions, respectively Scale bars: 10 Å (f) and 5 Å (g) 
Fig  3 | Characterization of closed-shell indeno[1,2-a]fluorene on bilayer NaCl/Au(111) a, DFTcalculated wave functions of the frontier orbitals of closed-shell 5 0 (isovalue: 0 002 e -Å -3 ) The wave functions shown here are calculated for the 5para geometry b, DFT-calculated bond lengths of 5ortho (top) and 5para (bottom) c, Constant-height I(V) spectra acquired on a species of 5 assigned as 5para, along with the corresponding dI/dV(V) spectra Open feedback parameters: V = -2 V, I = 0 15 pA (negative bias side) and V = 2 2 V, I = 0 15 pA (positive bias side) Acquisition position of the spectra is shown in Supplementary Fig  7  d, Scheme of many-body transitions associated to the measured ionic resonances of 5para Also shown are STM images of assigned 5para at biases where the corresponding transitions become accessible Scanning parameters: I = 0 15 pA (V = -1 5 V) and 0 2 pA (V = 1 7 V)  e, Laplace-filtered AFM image of assigned 5para STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 7 Å  f, Selected bonds labeled for highlighting bond order differences between 5para and 5ortho For the bond pairs a/b, c/d and e/f, the bonds labeled in bold exhibit a higher bond order than their neighboring labeled bonds in 5para g, Laplace-filtered AFM images of 5 on bilayer NaCl/Cu(111) showing switching between 5OS and 5para as the molecule changes its adsorption position The faint protrusion adjacent to 5 is a defect that stabilizes the adsorption of 5  STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3Å  STM and STS data in c and d are acquired on the same species, while the AFM data in e is acquired on a different species Scale bars: 10 Å (d) and 5 Å (e,g) 
NMR (300 MHz, CDCl3) δ: 7 51 (m, 2H), 7 40 -7 28 (m, 5H), 7 27 -7 20 (m, 2H), 7 13 (d, J = 7 7 Hz, 1H), 2 07 (s, 3H), 1 77 (s, 3H) ppm  13C NMR-DEPT (75 MHz, CDCl3, 1:1 mixture of atropisomers) δ: 141 2 (C), 141 1 (C), 140 0 (C), 139 4 (2C), 137 5 (C), 137 4 (C), 136 0 (3C), 134 8 (C), 134 5 (C), 134 1 (C), 134 0 (C), 133 7 (C), 133 6 (C), 131 6 (CH), 131 2 (CH), 131 1 (CH), 130 7 (CH), 129 8 (CH), 129 7 (CH), 129 5 (CH), 129 4 (CH), 129 0 (CH), 128 9 (CH), 128 7 (2CH), 128 6 (2CH), 127 2 (CH), 127 1 (CH), 127 0 (CH), 126 9 (CH), 126 7 (CH), 126 6 (CH), 20 6 (CH3), 20 5 (CH3), 17 7 (CH3), 17 5 (CH3) ppm MS (APCI) m/z (%): 327 (M+1, 100) HRMS: C20H16Cl2; calculated: 327 0702, found: 327 0709 
NMR (500 MHz, CDCl3) δ: 7 93 (d, J = 7 6 Hz, 1H), 7 85 (d, J = 7 5 Hz, 1H), 7 78 (d, J = 7 7 Hz, 1H), 7 65 (d, J = 7 4 Hz, 1H), 7 61 (d, J = 7 5 Hz, 1H), 7 59 (d, J = 7 7 Hz, 1H), 7 47 (ddd, J = 8 4, 7 2, 1 1 Hz, 1H), 7 42 (dd, J = 8 1, 7 0 Hz, 1H), 7 35 (m, 2H), 4 22 (s, 3H), 4 02 (s, 3H) ppm  13C NMR-DEPT (125 MHz, CDCl3) δ: 144 1 (C), 143 3 (C), 142 3 (C), 141 9 (C), 141 8 (C), 141 2 (C), 138 2 (C), 136 5 (C), 127 0 (CH), 126 9 (CH), 126 7 (CH), 126 6 (CH), 125 3 (CH), 125 2 (CH), 123 6 (CH), 122 2 (CH), 119 9 (CH), 118 4 (CH), 37 4 (CH2), 36 3 (CH2) ppm MS (APCI) m/z (%): 254 (M+, 88) HRMS: C20H14; calculated: 254 1090, found: 254 1090 

abstract

Indenofluorenes are non-benzenoid conjugated hydrocarbons that have received great interest owing to their unusual electronic structure and potential applications in nonlinear optics and photovoltaics  Here, we report the generation of unsubstituted indeno[1,2-a]fluorene, the final and yet unreported parent indenofluorene regioisomer, on various surfaces by cleavage of two C-H bonds in 7,12-dihydro indeno[1,2-a]fluorene through voltage pulses applied by the tip of a combined scanning tunneling microscope and atomic force microscope 
On bilayer NaCl on Au(111), indeno[1,2a]fluorene is in the neutral charge state, while it exhibits charge bistability between neutral and anionic states on the lower work function surfaces of bilayer NaCl on Ag(111) and Cu(111)  In the neutral state, indeno[1,2-a]fluorene exhibits either of two ground states: an open-shell π-diradical state, predicted to be a triplet by density functional and multireference many-body perturbation theory calculations, or a closedshell state with a para-quinodimethane moiety in the as-indacene core 
Switching between open-and closed-shell states of a single molecule is observed by changing its adsorption site on NaCl  The inclusion of non-benzenoid carbocyclic rings is a viable route to tune the physicochemical properties of polycyclic conjugated hydrocarbons (PCHs)   Non-benzenoid polycycles may lead to local changes in strain, conjugation, aromaticity, and, relevant to the context of the present work, induce an open-shell ground state of the corresponding PCHs  
Many nonbenzenoid PCHs are also non-alternant, where the presence of odd-membered polycycles breaks the bipartite symmetry of the molecular network   Figure shows classical examples of non-benzenoid non-alternant PCHs, namely, pentalene, azulene and heptalene  Whereas azulene is a stable PCH exhibiting Hückel aromaticity ([4n+2] π-electrons, n = 2), pentalene and heptalene are unstable Hückel antiaromatic compounds with [4n] π-electrons, n = 2 (pentalene) and n = 3 (heptalene) 
Benzinterposition of pentalene generates indacenes, consisting of two isomers s-indacene and as-indacene (Fig  )  Apart from being antiaromatic, indacenes also contain proaromatic quinodimethane (QDM) moieties (Fig  ) , which endows them with potential open-shell character  While the parent s-indacene and asindacene have never been isolated, stable derivatives of s-indacene bearing bulky substituents have been synthesized  
A feasible strategy to isolate congeners of otherwise unstable non-benzenoid non-alternant PCHs is through fusion of benzenoid rings at the ends of the π-system, that is, benzannelation  For example, while the parent pentalene is unstable, the benzannelated congener indeno[2,1-a]indene is stable under ambient conditions (Fig  )  
However, the position of benzannelation is crucial for stability: although indeno[2,1a]indene is stable, its regioisomer indeno[1,2-a]indene (Fig  ) oxidizes under ambient conditions   Similarly, benzannelation of indacenes gives rise to the family of PCHs known as indenofluorenes (Fig  ), which constitute the topic of the present work 
Depending on the benzannelation position and the indacene core, five regioisomers can be constructed, namely, indeno [ Practical interest in indenofluorenes stems from their low frontier orbital gap and excellent electrochemical characteristics that render them as useful components in organic electronic devices  
The potential open-shell character of indenofluorenes has led to several theoretical studies on their use as non-linear optical materials and as candidates for singlet fission in organic photovoltaics   Recent theoretical work has also shown that indenofluorene-based ladder polymers may exhibit fractionalized excitations 
Fundamentally, indenofluorenes represent model systems to study the interplay between aromaticity and magnetism at the molecular scale   Motivated by many of these prospects, the last decade has witnessed intensive synthetic efforts toward the realization of indenofluorenes  Derivatives of 1-4 have been realized in solution , while 1-3 have also been synthesized on surfaces and characterized using scanning tunneling microscopy (STM) and atomic force microscopy (AFM), which provide information on molecular orbital densities , molecular structure and oxidation state  
With regards to the open-shell character of indenofluorenes, 2-4 are theoretically and experimentally interpreted to be closed-shell, while calculations indicate that 1 and 5 should exhibit open-shell ground states   Bulk characterization of mesitylsubstituted 1, including X-ray crystallography, temperature-dependent NMR, and electron spin resonance spectroscopy, provided indications of its open-shell ground state  
Electronic characterization of 1 on Au(111) surface using scanning tunneling spectroscopy (STS) revealed a low electronic gap of 0 4 eV (ref  )  However, no experimental proof of an openshell ground state of 1 on Au(111), such as detection of singly occupied molecular orbitals (SOMOs) or spin excitations and correlations due to unpaired electrons , was shown 
In this work, we report the generation and characterization of unsubstituted 5  Our research is motivated by theoretical calculations that indicate 5 to exhibit the largest diradical character among all indenofluorene isomers   The same calculations also predict that 5 should possess a triplet ground state 
Therefore, 5 would qualify as a Kekulé triplet, of which only a handful of examples exist   However, definitive synthesis of 5 has never been reported so far  Previously, Dressler et al  reported transient isolation of mesityl-substituted 5, where it decomposed both in the solution and in solid state , and only the structural proof of the corresponding dianion was obtained 
On-surface generation of a derivative of 5, starting from truxene as a precursor, was recently reported   STM data on this compound, containing the indeno[1,2-a]fluorene moiety as part of a larger PCH, was interpreted to indicate its open-shell ground state  However, the results did not imply the ground state of unsubstituted 5  Here, we show that on insulating surfaces 5 can exhibit either of two ground states: an open-shell or a closed-shell 
We infer the existence of these two ground states based on high-resolution AFM imaging with bond-order discrimination and STM imaging of molecular orbital densities   AFM imaging reveals molecules with two different geometries  Characteristic bond-order differences in the two geometries concur with the geometry of either an open-or a closed-shell state 
Concurrently, STM images at ionic resonances show molecular orbital densities corresponding to SOMOs for the open-shell geometry, but orbital densities of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for the closed-shell geometry  Our experimental results are in good agreement with density functional theory (DFT) and multireference perturbation theory calculations 
Finally, we observe switching between open-and closed-shell states of a single molecule by changing its adsorption site on the surface  Synthetic strategy toward indeno[1,2-a]fluorene  The generation of 5 relies on the solution-phase synthesis of the precursor 7,12-dihydro indeno[1,2-a]fluorene (6)  Details on synthesis and characterization of 6 are reported in Supplementary Figs 
  Single molecules of 6 are deposited on coinage metal (Au(111), Ag(111) and Cu(111)) or insulator surfaces  In our work, insulating surfaces correspond to two monolayer-thick (denoted as bilayer) NaCl on coinage metal surfaces
Context 3: Paper Info

Title: Interpretable reduced-order modeling with time-scale separation Interpretable reduced-order modeling with time-scale separation
Publish Date: 7 March 2023
Author List: Sebastian Kaltenbach (from CSE-Lab, ETH Zurich, Harvard SEAS), Phaedon-Stelios Koutsourelakis (from CSE-Lab, ETH Zurich, Harvard SEAS), Petros Koumoutsakos (from CSE-Lab, ETH Zurich, Harvard SEAS), Harvard Seas (from CSE-Lab, ETH Zurich, Harvard SEAS)

Figure

FIG  5  Comparison between the phase-space of the reference solution (left) and the phase-space of the predictions
FIG  7  Comparison between predictions and reference solutions for a new initial condition fort = 1 25, 3 75, 7 5, 12 5, 20, 30  (from left to right and top to down) We note that with longer prediction time the uncertainty bounds increases Despite the chaotic nature of the KS equation, the predictive posterior mean is close to the reference solution for t ≤ 12 5

abstract

Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering  However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial  We propose such a data-driven scheme that automates the identification of the time-scales involved and, can produce stable predictions forward in time as well as under different initial conditions not included in the training data 
To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space  It readily allows for the inclusion of sparse and irregularly sampled training data  The learned, latent dynamics are interpretable and reveal the different temporal scales involved 
We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix  Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation 
Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework 

INTRODUCTION

High-fidelity simulations of critical phenomena such as ocean dynamics and epidemics have become essential for decision-making  They are based on physically-motivated PDEs expressing system dynamics that span multiple spatiotemporal scales and which necessitate cumbersome computations   In recent years there is increased attention to the development of data-driven models that can accelerate the solution of these PDEs as well as reveal salient, lower-dimensional features that control the long-term evolution 
In most cases, data-driven reduced-order models are not interpretable  In particular, models based on neural networks despite good predictive capabilities , they offer a black-box description of the system dynamics  A possible remedy is applying a symbolic regression to the learned neural network representation , but this adds additional computational cost due to the two-step procedure 
A number of frameworks such as SINDy allows to learn interpretable dynamics but it relies on the a-priori availability of lower-dimensional descriptors and of time-derivatives which can be very noisy for both simulation and experimental data  Other frameworks are tailored to specific problems such as molecular dynamics  
Here, we present a framework that only needs the value of the observables, and not their derivatives, as training data and is capable of identifying interpretable latent dynamics  The deployment of interpretable latent dynamics ensures that conservation of important properties of that are reflected in the reduced-order model  
The present method is related to approaches based on the Koopman-operator extended Dynamic Mode Decomposition (eDMD) but uses continuous complex-valued latent space dynamics and only requires one scalar variable per latent dimension to describe the latent space dynamics  Therefore we do not have to enforce any parametrizations on the Koopman matrix  
The time-continuous formulation moreover allows to incorporate sparse and irregularly sampled training data and fast generation of predictions after the training phase  By using a complex-valued latent space we can also incorporate harmonic effects and reduce the number of latent variables needed  Linear and non-linear autoencoders are used to map the observed, high-dimensional time-series to the lower-dimensional, latent representation and we identify simultaneously the autoencoder as well as the latent dynamics by optimizing a combined loss function 
Hence the to tasks of dimensionality reduction and discovery of the reduced dynamics are unified while other frameworks treat the two parts separately   Apart from using an architecture based on autoencoders to identify the latent space, projection-based methods could also be employed   We are also proposing a probabilistic version of our algorithm ) that makes use of probabilistic Slow Feature Analysis  
This allows for a latent representation that arart from being time-continuous, can quantify the predictive uncertainty and hierarchically decompose the dynamics into their pertinent scales while promoting the discovery of slow processes that control the system's evolution over long time horizons  The rest of the paper is structured as follows: We introduce the methodological framework as well as algorithmic details in section II 
Particular focus is paid on the interpretability of the inferred lower-dimensional dynamics  In section III we present three numerical illustrations, i e  a system of linear ODEs, a hidden Markov Model and the discretized KS-equation  We then present in section IV the probabilistic extension of the framework and apply it to the KS-equation 
We conclude with a summary and a short discussion about possible next steps  We introduce the autoencoders deployed in this work, followed by the interpretable latent space dynamic and discuss the training process  We consider data from high-dimensional time series x n ∈ R f with n = 1,   



ground_truth: Quill harmed states more than anticipated due to the Internet.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Can individual molecules of indeno[1,2-a]fluorene switch between open-shell and closed-shell states?

Read the following text and answer briefly.

Context 2:  Voltage pulses ranging between 4-6 V are applied by the tip of a combined STM/AFM system, which result in cleavage of one C-H bond at each of the pentagonal apices of 6, thereby leading to the generation of 5 (Fig  ) 
In the main text, we focus on the generation and characterization of 5 on insulating surfaces  Generation and characterization of 5 on coinage metal surfaces is shown in Supplementary Fig    )  Blue and orange colors represent spin up and spin down densities, respectively  c, Probability density of the SOMOs of 5OS (isovalue: 0 001 e -Å -3 ) 
d, DFT-calculated bond lengths of 5OS  e, Constant-height I(V) spectra acquired on a species of 5 assigned as 5OS, along with the corresponding dI/dV(V) spectra  Open feedback parameters: V = -2 V, I = 0 17 pA (negative bias side) and V = 2 V, I = 0 17 pA (positive bias side)  Acquisition position of the spectra is shown in Supplementary Fig    f, Scheme of many-body transitions associated to the measured ionic resonances of 5OS 
Also shown are STM images of assigned 5OS at biases where the corresponding transitions become accessible  Scanning parameters: I = 0 3 pA (V = -1 2 V and -1 5 V) and 0 2 pA (V = 1 3 V and 1 6 V)  g, Laplace-filtered AFM image of assigned 5OS  STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3
Å  The tip-height offset Δz for each panel is provided with respect to the STM setpoint, and positive (negative) values of Δz denote tip approach (retraction) from the STM setpoint  f and g show the same molecule at the same adsorption site, which is next to a trilayer NaCl island  The bright and dark features in the trilayer NaCl island in g correspond to Cl -and Na + ions, respectively 
Scale bars: 10 Å (f) and 5 Å (g)  To experimentally explore the electronic structure of 5, we used bilayer NaCl films on coinage metal surfaces to electronically decouple the molecule from the metal surfaces  Before presenting the experimental findings, we summarize the results of our theoretical calculations performed on 5 in the neutral charge state (denoted as 5 0 ) 
We start by performing DFT calculations on 5 0 in the gas phase  Geometry optimization performed at the spin-unrestricted UB3LYP/6-31G level of theory leads to one local minimum, 5OS, the geometry of which corresponds to the open-shell resonance structure of 5 (Fig  , the label OS denotes open-shell) 
The triplet electronic configuration of 5OS is the lowest-energy state, with the openshell singlet configuration 90 meV higher in energy  Geometry optimization performed at the restricted closed-shell RB3LYP/6-31G level reveals two local minima, 5para and 5ortho, the geometries of which (Fig  ) exhibit bond length alternations in line with the presence of a para-or an ortho-QDM moiety, respectively, in the as-indacene core of the closed-shell resonance structures of 5 (Fig  )  
Relative to 5OS in the triplet configuration, 5para and 5ortho are 0 40 and 0 43 eV higher in energy, respectively  Additional DFT results are shown in Supplementary Fig    To gain more accurate insights into the theoretical electronic structure of 5, we performed multireference perturbation theory calculations (Supplementary Fig  ) based on quasi-degenerate second-order n-electron valence state perturbation theory (QD-NEVPT2) 
In so far as the order of the ground and excited states are concerned, the results of QD-NEVPT2 calculations qualitatively match with DFT calculations  For 5OS, the triplet configuration remains the lowest-energy state, with the open-shell singlet configuration 60 meV higher in energy  The energy differences between the open-and closed-shell states are substantially reduced in QD-NEVPT2 calculations, with 5para and 5ortho only 0 11 and 0 21 eV higher in energy, respectively, compared to 5OS in the triplet configuration 
We also performed nucleus-independent chemical shift calculations to probe local aromaticity of 5 in the openand closed-shell states  While 5OS in the triplet configuration exhibits local aromaticity at the terminal benzenoid rings, 5OS in the open-shell singlet configuration, 5para and 5ortho all display antiaromaticity (Supplementary Fig  ) 
The choice of the insulating surface determines the charge state of 5: while 5 adopts neutral charge state on the high work function bilayer NaCl/Au(111) surface (irrespective of its openor closed-shell state, Supplementary Fig  ), 5 exhibits charge bistability between 5 0 and the anionic state 5 -1 on the lower work function bilayer NaCl/Ag(111) and Cu(111) surfaces (Supplementary Figs  ) 
In the main text, we focus on the characterization of 5 on bilayer NaCl/Au(111)  Characterization of charge bistable 5 is reported in Supplementary Figs    We first describe experiments on 5 on bilayer NaCl/Au(111), where 5 exhibits a geometry corresponding to the calculated 5OS geometry, and an open-shell electronic configuration 
We compare the experimental data on this species to calculations on 5OS with a triplet configuration, as theory predicts a triplet ground state for 5OS  For 5OS, the calculated frontier orbitals correspond to the SOMOs ψ1 and ψ2 (Fig  ), whose spin up levels are occupied and the spin down levels are empty 
Figure shows the DFT-calculated bond lengths of 5OS, where the two salient features, namely, the small difference in the bond lengths within each ring and the notably longer bond lengths in the pentagonal rings, agree with the open-shell resonance structure of 5 (Fig  )  Figure shows an AFM image of 5 adsorbed on bilayer NaCl/Au(111) that we assign as 5OS, where the bond-order differences qualitatively correspond to the calculated 5OS geometry (discussed and compared to the closed-shell state below) 
Differential conductance spectra (dI/dV(V), where I and V denote the tunneling current and bias voltage, respectively) acquired on assigned 5OS exhibit two peaks centered at -1 5 V and 1 6 V (Fig  ), which we assign to the positive and negative ion resonances (PIR and NIR), respectively  Figure shows the corresponding STM images acquired at the onset (V = -1 2
V/1 3 V) and the peak (V = -1 5 V/1 6 V) of the ionic resonances  To draw a correspondence between the STM images and the molecular orbital densities, we consider tunneling events as many-body electronic transitions between different charge states of 5OS (Fig  )  Within this framework, the PIR corresponds to transitions between 5 0 and the cationic state 5  
At the onset of the PIR at -1 2 V, an electron can only be detached from the SOMO ψ1 and the corresponding STM image at -1 2 V shows the orbital density of ψ1  Increasing the bias to the peak of the PIR at -1 5 V, it becomes possible to also empty the SOMO ψ2, such that the corresponding STM image shows the superposition of ψ1 and ψ2, that is, |ψ1| 2 + |ψ2| 2 (ref 
)  Similarly, the NIR corresponds to transitions between 5 0 and 5 -1   At the NIR onset of 1 3 V, only electron attachment to ψ2 is energetically possible  At 1 6 V, electron attachment to ψ1 also becomes possible, and the corresponding STM image shows the superposition of ψ1 and ψ2  The observation of the orbital densities of SOMOs, and not the hybridized HOMO and LUMO, proves the open-shell ground state of assigned 5OS 
Measurements of the monoradical species with a doublet ground state are shown in Supplementary Fig    Unexpectedly, another species of 5 was also experimentally observed that exhibited a closedshell ground state  In contrast to 5OS, where the frontier orbitals correspond to the SOMOs ψ1 and ψ2, DFT calculations predict orbitals of different shapes and symmetries for 5para and 5ortho, denoted as α and β and shown in Fig   
For 5ortho, α and β correspond to HOMO and LUMO, respectively  The orbitals are inverted in energy and occupation for 5para, where β is the HOMO and α is the LUMO  Fig  shows an AFM image of 5 that we assign as 5para  We experimentally infer its closed-shell state first by using qualitative bond order discrimination by AFM 
In high-resolution AFM imaging, chemical bonds with higher bond order are imaged brighter (that is, with higher frequency shift Δf) due to stronger repulsive forces, and they appear shorter   In Fig  , we label seven bonds whose bond orders show significant qualitative differences in the calculated 5ortho, 5para (Fig  ) and 5OS (Fig  ) geometries 
In 5para, the bonds b and d exhibit a higher bond order than a and c, respectively  This pattern is reversed for 5ortho, while the bond orders of the bonds a-d are all similar and small for 5OS  Furthermore, in 5para bond f exhibits a higher bond order than e, while in 5ortho and 5OS bonds e and f exhibit similar bond order (because they belong to Clar sextets) 
Finally, the bond labeled g shows a higher bond order in 5para than in 5ortho and 5OS  The AFM image of assigned 5para shown in Fig  indicates higher bond orders of the bonds b, d and f compared to a, c and e, respectively  In addition, the bond g appears almost point-like and with enhanced Δf contrast compared to its neighboring bonds, indicative of a high bond order (see Supplementary Fig  for height-dependent measurements) 
These observations concur with the calculated 5para geometry (Fig  )  Importantly, all these distinguishing bond-order differences are distinctly different in the AFM image of 5OS shown in Fig  , which is consistent with the calculated 5OS geometry (Fig  )  In the AFM images of 5OS (Fig  and Supplementary Fig  ), the bonds a-d at the pentagon apices appear with similar contrast and apparent bond length 
The bonds e and f at one of the terminal benzenoid rings also exhibit similar contrast and apparent bond length, while the central bond g appears longer compared to assigned 5para  Further compelling evidence for the closed-shell state of assigned 5para is obtained by STM and STS  dI/dV(V) spectra acquired on an assigned 5para species exhibit two peaks centered at -1 4 V (PIR) and 1 6 V (NIR) (Fig  ) 
STM images acquired at these biases (Fig  ) show the orbital densities of β (-1 4 V) and α (1 6 V)  First, the observation of α and β as the frontier orbitals of this species, and not the SOMOs, strongly indicates its closed-shell state  Second, consistent with AFM measurements that indicate good correspondence to the calculated 5para geometry, we observe β as the HOMO and α as the LUMO 
For 5ortho, α should be observed as the HOMO and β as the LUMO  We did not observe molecules with the signatures of 5ortho in our experiments  We observed molecules in open-(5OS, Fig  ) and closed-shell (5para, Fig  ) states in similar occurrence after their generation from 6 on the surface  We could also switch individual molecules between open-and closed-shell states as shown in Fig  and Supplementary Fig   
To this end, a change in the adsorption site of a molecule was induced by STM imaging at ionic resonances, which often resulted in movement of the molecule  The example presented in Fig  shows a molecule that was switched from 5para to 5OS and back to 5para  The switching is not directed, that is, we cannot choose which of the two species will be formed when changing the adsorption site, and we observed 5OS and 5para in approximately equal yields upon changing the adsorption site 
The molecule in Fig  is adsorbed on top of a defect that stabilizes its adsorption geometry on bilayer NaCl  At defect-free adsorption sites on bilayer NaCl, that is, without a third layer NaCl island or atomic defects in the vicinity of the molecule, 5 could be stably imaged neither by AFM nor by STM at ionic resonances (Supplementary Fig  ) 
Without changing the adsorption site, the state of 5 (open-or closedshell) never changed, including the experiments on bilayer NaCl/Ag(111) and Cu(111), on which the charge state of 5 could be switched (Supplementary Figs  )  Also on these lower work function surfaces, both open-and closed-shell species were observed for 5 0 and both showed charge bistability between 5 0 (5OS or 5para) and 5 -1 (Supplementary Figs  ) 
The geometrical structure of 5 -1 probed by AFM, and its electronic structure probed by STM imaging at the NIR (corresponding to transitions between 5 -1 and the dianionic state 5 -2 ), are identical within the measurement accuracy for the charged species of both 5OS and 5para  When cycling the charge state of 5 between 5 0 and 5 -1 several times, we always observed the same state (5OS or 5para) when returning to 5 0 , provided the molecule did not move during the charging/discharging process 
Based on our experimental observations we conclude that indeno[1,2-a]fluorene (5), the last unknown indenofluorene isomer, can be stabilized in and switched between an open-shell (5OS) and a closed-shell (5para) state on NaCl  For the former, both DFT and QD-NEVPT2 calculations predict a triplet electronic configuration 
Therefore, 5 can be considered to exhibit the spin-crossover effect, involving magnetic switching between high-spin (5OS) and low-spin (5para) states, coupled with a reversible structural transformation  So far, the spin-crossover effect has mainly only been observed in transition-metal-based coordination compounds with a near-octahedral geometry  
The observation that the switching between open-and closedshell states is related to changes in the adsorption site but is not achieved by charge-state cycling alone, indicates that the NaCl surface and local defects facilitate different electronic configurations of 5 depending on the adsorption site 
Gas-phase QD-NEVPT2 calculations predict that 5OS is the ground state, and the closed-shell 5para and 5ortho states are 0 11 and 0 21 eV higher in energy  The experiments, showing bidirectional switching between 5OS and 5para, indicate that a change in the adsorption site can induce sufficient change in the geometry of 5 (leading to a corresponding change in the ground state electronic configuration) and thus induce switching 
Switching between open-and closed-shell states in 5 does not require the breaking or formation of covalent bonds , but a change of adsorption site on NaCl where the molecule is physisorbed  Our results should have implications for single-molecule devices, capitalizing on the altered electronic and chemical properties of a system in π-diradical open-shell and closed-shell states such as frontier orbital and singlet-triplet gaps, and chemical reactivity 
For possible future applications as a single-molecule switch, it might be possible to also switch between open-and closed-shell states by changing the local electric field, such as by using chargeable adsorbates   Scanning probe microscopy measurements and sample preparation  STM and AFM measurements were performed in a home-built system operating at base pressures below 1×10 -10 mbar and a base temperature of 5 K  Bias voltages are provided with respect to the sample 
All STM, AFM and spectroscopy measurements were performed with carbon monoxide (CO) functionalized tips  AFM measurements were performed in non-contact mode with a qPlus sensor   The sensor was operated in frequency modulation mode with a constant oscillation amplitude of 0 5 Å  STM measurements were performed in constantcurrent mode, AFM measurements were performed in constant-height mode with V = 0 V, and I(V) and Δf(V) spectra were acquired in constant-height mode 
Positive (negative) values of the tip-height offset Δz represent tip approach (retraction) from the STM setpoint  All dI/dV(V) spectra are obtained by numerical differentiation of the corresponding I(V) spectra  STM and AFM images, and spectroscopy curves, were post-processed using Gaussian low-pass filters 
Au(111), Ag(111) and Cu(111) surfaces were cleaned by iterative cycles of sputtering with Ne + ions and annealing up to 800 K  NaCl was thermally evaporated on Au(111), Ag(111) and Cu(111) surfaces held at 323 K, 303 K and 283 K, respectively  This protocol results in the growth of predominantly bilayer (100)-terminated islands, with a minority of trilayer islands 
Sub-monolayer coverage of 6 on surfaces was obtained by flashing an oxidized silicon wafer containing the precursor molecules in front of the cold sample in the microscope  CO molecules for tip functionalization were dosed from the gas phase on the cold sample  Density functional theory calculations  DFT was employed using the PSI4 program package  
All molecules with different charge (neutral and anionic) and electronic (open-and closed-shell) states were independently investigated in the gas phase  The B3LYP exchangecorrelation functional with 6-31G basis set was employed for structural relaxation and singlepoint energy calculations  The convergence criteria were set to 10 −4 eV Å −1 for the total forces and 10 −6 eV for the total energies 
Multireference calculations  Multireference calculations were performed on the DFToptimized geometries using the QD-NEVPT2 level of theory , with three singlet roots and one triplet root included in the state-averaged calculation  A (10,10) active space (that is, 10 electrons in 10 orbitals) was used along with the def2-TZVP basis set  
Increasing either the active space size or expanding the basis set resulted in changes of about 50 meV for relative energies of the singlet and triplet states  These calculations were performed using the ORCA program package   Nucleus-independent chemical shift (NICS) calculations  Isotropic nucleus-independent chemical shift values were evaluated at the centre of each ring using the B3LYP exchangecorrelation functional with def2-TZVP basis set using the Gaussian 16 software package  
Starting materials (reagent grade) were purchased from TCI and Sigma-Aldrich and used without further purification  Reactions were carried out in flame-dried glassware and under an inert atmosphere of purified Ar using Schlenk techniques  Thin-layer chromatography (TLC) was performed on Silica Gel 60 F-254 plates (Merck) 
Column chromatography was performed on silica gel (40-60 µm)  Nuclear magnetic resonance (NMR) spectra were recorded on a Bruker Varian Mercury 300 or Bruker Varian Inova 500 spectrometers  Mass spectrometry (MS) data were recorded in a Bruker Micro-TOF spectrometer  The synthesis of compound 6 was developed following the two-step synthetic route shown in Supplementary Fig  , which is based on the preparation of methylene-bridge polyarenes by means of Pd-catalyzed activation of benzylic C-H bonds  
Supplementary Figure | Synthetic route to obtain compound 6  The complex Pd2(dba)3 (20 mg, 0 02 mmol) was added over a deoxygenated mixture of 1,3-dibromo-2,4-dimethylbenzene (9, 100 mg, 0 38 mmol), boronic acid 10 (178 mg, 1 14 mmol), K2CO3 (314 mg, 2 28 mmol) and XPhos (35 mg, 0 08 mmol) in toluene (1:1, 10 mL), and the resulting mixture was heated at 90 °C for 2 h 
After cooling to room temperature, the solvents were evaporated under reduced pressure  The reaction crude was purified by column chromatography (SiO2; hexane:CH2Cl2 9:1) affording 11 (94 mg, 76%) as a colorless oil  The complex Pd(OAc)2 (7 mg, 0 03 mmol) was added over a deoxygenated mixture of terphenyl 11 (90 mg, 0 27 mmol), K2CO3 (114 mg, 0 83 mmol) and ligand L (26 mg, 0 06 mmol) in NMP (2 mL) 
The resulting mixture was heated at 160 °C for 4 h  After cooling to room temperature, H2O (30 mL) was added, and the mixture was extracted with EtOAc (3x15 mL)  The combined organic extracts were dried over anhydrous Na2SO4, filtered, and evaporated under reduced pressure  The reaction crude was purified by column chromatography (SiO2; hexane:CH2Cl2 9:1) affording compound 6 (8 mg, 11%) as a white solid  in AFM imaging due to their reduced adsorption height compared to the rest of the carbon atoms 
We attribute this observation to the significantly different lattice parameter of Cu(111) (2 57 Å) compared to Au(111) and Ag(111) (2 95 Å and 2 94 Å, respectively) , such that the apical carbon atoms of the pentagonal rings of 5 adsorb on the on-top atomic sites on Au(111) and Ag(111), but not on Cu(111) 
Our speculation is based on a previous study of polymers of 1 on Au(111) by Di Giovannantonio et al  , where both tilted and planar individual units of 1 were observed depending on whether the apical carbon atoms of the pentagonal rings in 1 adsorbed on the on-top or hollow sites of the surface, respectively 
Given the strong molecule-metal interaction, we found no electronic state signatures of 5 on all three metal surfaces  STM set point for AFM images: V = 0  e, Frontier orbital spectrum of 5 -1   In the anionic state, ψ2 becomes doubly occupied and ψ1 is the SOMO  Filled and empty circles denote occupied and empty orbitals, respectively 
For each panel, zero of the energy axis has been aligned to the respective highest-energy occupied orbital 
Context 3: , T   We remark that the intervals between the different states do not need to be uniformly spaced 

Autoencoder

A core assumption of the method is that each high-dimensional state x n can be compressed to a lower-dimensional representation z n ∈ C c with c << f   We identify this lower-dimensional representation by an autoencoder consisiting of a parameterized encoder and decoder  The encoder maps the high-dimensional representation to the latent space as:
The latent space is complex-valued  The decoder reconstructs the high-dimensional representation based on the latent variables as: We denote the parameters of the encoder as well as the decoder by θ  As discussed later in Section II C, both set of parameters are optimized simultaneously during training and therefore there is no need for differentiating them 

Interpretable Latent Space Dynamics

We employ a propagator in the latent space to capture the reduced-order dynamics of the system  In contrast to other time-extended variational autoencoder frameworks, our representation uses complex valued latent variables  In addition the latent variables are treated independently  The latter feature enables us to have an interpretable latent dynamics as well as a model that is especially suitable for being trained in the Small Data regime due to the small number of required parameters 
This is in contrast to temporal propagators such as LSTMs   For each dimension i of the latent variable z we are using the following continuous ODE in the complex plane: By solving this ODE, we can define the operator: Interpretable reduced-order modeling with time-scale separation Here, λ is a vector containing all the individual λ's and ∆t n indicates the time-step between the latent states 
The symbol is used to indicate a component-wise multiplication  We remark that the latent variables and the parameter governing the temporal evolution are complex numbers and their role in describing the system dynamics is similar to that of an eigenvalue  The real part is associated with growth and decay whereas the imaginary part is representing the periodic component 
This approach has similarities with the Koopman-operator based methods and the extended dynamic mode decomposition   In contrast to the methods mentioned before we are using a continuous formulation in the latent space that allows us to incorporate scarce and irregularly sampled training data and directly rely on complex numbers in the latent space 

Training and Predictions

We optimize a loss function that combines both a reconstruction loss as well as a loss associated with the error of our learned propagator in the latent space: (5) We note that we could directly incorporate mini-batch training by only taking the summation over a subset of the N available training data 
For new predictions of unseen states, we use the encoder to generate a latent representations which is then advanced in time by the learned propagator  At a designated time step we are using the decoder to reconstruct the high-dimensional solution  We applied our algorithm to three systems  First, we show that the algorithm is capable of exactly reproducing the solution of a linear ODE and to identify its eigenvalues 
Afterwards we are applying the framework to a high-dimensional process generated by a complex latent dynamics, which is correctly identified  As a final test case, we are applying the algorithm to a Kuramoto Shivashinski (KS) equation  Interpretable reduced-order modeling with time-scale separation

Linear ODE

We are considering a two-dimensional ODE system for x = y 1 y 2 : Based on the obtained training data we run our algorithm using a linear encoder and decoder structure as well as two latent variables z  The loss function was optimized using the Adam algorithm   As we consider a linear ODE we can analytically compute the eigenvalues involved and compare it with the parameters λ identified by our algorithm 
We observe in Figure that the algorithm was able to recover the correct values, i e  the eigenvalues 7 and 3 of the given linear ODE  The system does not have a periodic component and the two imaginary parts correctly go to zero, whereas the real parts converge to the reference value  Moreover we are also able to identify for the linear mapping between our latent variables z and the training data a matrix consisting of a multiple of the eigenvectors (1,1) and (1,-1) and thus the correct solution 
This example was chosen to show that the algorithm is able to quickly identify the exact solution of a linear ODE in terms of its linearly independent components 

Hidden multiscale dynamics

We consider eight-dimensional synthetic time series data produced by an underlying twodimensional complex valued process  In particular, the data points x are generated by first solving for the temporal evolution for the two complex-valued processes p 1 and p 2 and than mapping to the eight-dimensional space by using a randomly sampled linear mapping W  
One of the two processes used to generate the data is chosen to be much slower than the other one and both processes have a periodic component  dp 2 dt = (−0 9 + 1 5i)p 2 (8) As training data we consider 40 time series with 150 data points each, obtained by simulating the described processes for a maximum of t = 15 s and then sampling from the obtained data points 
Hence the training data consists of: • 40 time-series • with each consisting 150 observations of the x at a uniform time-step ∆t = 0 0025 The autoencoder obtained consists of one linear layer for both the decoder as well as the encoder  The model is trained for 5000 iterations using the Adam optimizer and a learning rate of 10 −3  
The results for the convergence of the parameters λ 1 and λ 2 can be found in Figure   We note that the process which is slower decaying and thus more responsible for the long-term evolution of the system has a higher convergence rate than the faster process  With the obtained parameters λ as well as the trained autoencoder, we compute predictions based on the last time step used for training, i e  we apply the encoder to obtain our latent representation and than use the latent dynamics to advance the latent representation in time 
Afterwards, we employ the decoder to reconstruct the full high-dimensional system  The results can be found in Figure and show very good agreement between predictions and reference data



ground_truth: Yes, individual molecules of indeno[1,2-a]fluorene can switch between open-shell and closed-shell states by changing their adsorption site on the surface.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: When was the paper published?

Read the following text and answer briefly.

Context 3:  This example shows that our model is successfully able to carry out dimensionality reduction and moreover indicates that the convergence rate between latent processes can be different 
The latter is relevant when training models as for accurate predictions all latent processes and their dynamics should be converged 

Kuramoto-Sivashinsky

Finally, we applied our algorithm to the KS equation and aim to identify a reduced-order model for the solution u(y, t): We employed periodic boundary conditions, µ = 1 and a domain size y ∈ [0, 22]  For this domain-size, the KS-equation exhibits a structurally stable chaotic attractor as discussed in The black lines divides the area for which training data was given from the area without raining data 
;   The equation is discretized in space using a discretization step of 22 64 resulting in a state vector x of dimension 64 and a nonlinear system of coupled ODEs  This is solved using a stiff fourth-order solver  We employed a non-linear encoder and decoder with four fully-connected layers each and ReLU-activation functions as well as Dropout Layers between the fully-connected layers 
We trained the model for 200000 iterations using Adam and a learning rate of 5 • 10 4 and assuming a five-dimensional latent space  We obtained the λ's in Figure   Four latent variables have λ's close to zero and thus a slow temporal dynamic that is responsible for the long-term evolution whereas one latent variable is quickly decaying 
Based on the obtained parameters, we do predictions based on an unseen initial condition not contained in the training data  We are able to reconstruct the correct phase space based on our predictions despite only using a very limited amount of training data  The results for the phase space can be seen in Figure  
Although the small-scale fluctuations in the temporal dynamics are not well captured, the model identifies the correct manifold which has a good accuracy compared to the reference solution  All phase-spaces were obtained by using a finite-difference operator on the data or predictions  These results are in accordance Interpretable reduced-order modeling with time-scale separation with whose LSTM-based temporal dynamic model was also able to find the correct phase space but not to track the actual dynamics for long-term predictions 
Our model is not able to account for noise in the temporal evolution and thus dealing with chaotic, small-scale fluctuations is challenging  We believe that a probabilistic version of our algorithm could be advantageous here  This section contains a fully probabilistic formulation for the deterministic model discussed before 
We replace the Autoencoder with a Variational Autoencoder and the ODE in the latent space with a SDE  The loss function which we optimize is the Evidence-Lower Bound (ELBO) 

Model Structure

We postulate the following relations for our probabilistic model using an Ornstein-Uhlenbeck (OU) for each dimension i of the latent space and a Wiener process W t in the latent space: We again assume that the latent variables z t are complex-valued and a priori independent  Complex variables were chosen as their evolution includes a harmonic components which are observed in many physical systems 
We assume an initial conditions z 0,i ∼ CN (0, σ 2 0,i )  The total parameters associated with the latent space dynamics of our model are thus {σ 2 0,i , σ 2 i , λ i } c i=1 and will be denoted by θ together with all parameters responsible for the decoder mapping G (see next section)  These parameters along with the state variables z t have to be inferred from the data x t  
Based on probabilistic Slow Feature Analysis (SFA) , we set σ 2 i = 2; (λ j ) and σ 2 0,i = 1  As a consequence, a priori, the latent dynamics are stationary  A derivation and reasoning for this choice can be found in Appendix A  Hence the only independent parameters are the λ i , the imaginary part of which can account for periodic effects in the latent dynamics 

Variational Autoencoder

We employ a variational autoencoder to account for a probabilistic mappings from the lower-dimensional representation z n to the high-dimensional system x n   In particular we are employing a probabilistic decoder The encoder is used to infer the state variables z based on the given data and thus defined in the inference and learning section 

Inference and Learning

Given the probabilistic relations , our goal is to infer the latent variables z 0:T as well as all model parameters θ  We follow a hybrid Bayesian approach in which the posterior of the state variables is approximated using amortized Variational Inference and Maximum-A-Posteriori (MAP) point-estimates for θ are computed 
The application of Bayes' rule for each data sequence x 0:T leads to the following posterior: where p(θ) denotes the prior on the model parameters  In the context of variational inference, we use the following factorization of the approximate posterior i e  we infer only the mean µ and variance σ for each state variable based on the given data points 
This conditional density used for inference is the encoder-counterpart to the probabilistic decoder defined in the section before  It can be readily shown that the optimal parameter values are found by maximizing the Evidence Lower Bound (ELBO) F(q φ (z 0:T ), θ) which is derived in Appendix B  We compute Monte Carlo estimates of the gradient of the ELBO with respect to φ and θ with the help of the reparametrization trick and carry out stochastic optimization with the ADAM algorithm  

Results for the probabilistic extension

We applied our probabilistic version to the KS-equation  We used the same settings as for the deterministic approach but considered up to 10 complex latent variables  The obtained λ's are in Figure   The probabilistic model allows us to quantify the uncertainty in predictions  In Figure predictions for various time-steps and the respective uncertainty bounds are shown for an unseen initial condition 
Due to the chaotic nature of the KS-equation and the small amount of training data, the underlying linear dynamic of our model is only able to capture the full dynamics for a limited time horizon  Fortunately, due to the probabilistic approach the model is capable of capturing chaotic fluctuations with increasingly wide uncertainty bounds 
We also computed the phase space representation for the KS-equation based on the predictions obtained by our model and compare it with the reference solution  The probabilistic model identifies the correct manifold with a better accuracy than the deterministic model  As some of the small-scale fluctuations are accounted as noise, the resulting manifold is more concentrated at the origin and the obtained values are slightly smaller than the reference manifold although their shape is very similar 



ground_truth: The paper was published on 7 March 2023.