The required mental state for the offense

Negligence alone is not willfulness. The law is clear on that.
The non-tax law does have a concept called gross negligence which is not just negligence. Even gross negligence is not willfulness for tax purposes unless it can be characterized as willful blindness.
The statutory term "willfully" is a chameleon, what the Supreme Court has called "a word of many meanings whose construction is often dependent on the context in which it appears.

 - taking the position that showing willful action requires proof only that a defendant had an "illegitimate or improper intent" to obstruct
 - "willfulness" is shown only when a defendant knew his conduct was unlawful at the time he engaged in it.
 -  read the term "willfully" to require some level of intent greater than "inadvertence or mere negligence".
-  the willfulness requirement of § 1701 requires proof that a defendant was aware of the specific provision violated.
 
In the context of criminal statutes, the word "willful" generally indicates a requirement of specific intent.... a “willful” act is one undertaken with a “bad purpose.” Said otherwise, in order to establish a “willful” violation of a statute, the Government must prove that the defendant acted with knowledge that his conduct was unlawful.
 The burden of proof for FBAR willfulness should be clear and convincing for the same reasons as for civil fraud.
The argument on the FBAR willful penalty would be like this:  (i) willfulness is required for the FBAR willful penalty; all authorities on the issue (including the IRM) interpret willfulness in the FBAR statute to be the same as the Cheek standard - intentional violation of a known legal duty (see e.g., Ratzlaf); this is the mens rea element for tax crimes (e.g., Cheek), the civil analogs of which (Section 6663) require proof by clear and convincing evidence; therefore, by the type of extension in Carlson, the FBAR willful civil penalty requires proof by clear and convincing evidence.
 “Willfulness” is a voluntary, intentional violation of a known legal duty.  A good-faith misunderstanding of the law, whether or not objectively reasonable, negates willfulness.  Cheek v. United States, 498 U.S. 192, 200-202, 112 L. Ed. 2d 617, 11 S. Ct. 604 (1991); United States v. Bishop, 412 U.S. 346, 360, 36 L. Ed. 2d 941, 93 S. Ct. 2009 (1973); United States v. Pensyl, 387 F.3d 456, 458-60 (6th Cir. 2004).  In United States v. Bishop, the Supreme Court observed:
In our complex tax system, uncertainty often arises even among taxpayers who earnestly wish to follow the law.  .  .  .  “It is not the purpose of the law to penalize frank difference of opinion or innocent errors made despite the exercise of reasonable care.”
412 U.S. at 360-61, quoting Spies v. United States, 317 U.S. 492, 496, 87 L. Ed 418, 63 S. Ct. 364 (1943).  Willfulness must include an evil motive and want of justification in view of all the circumstances.  James v. United States, 366 U.S. 213, 221, 6 L. Ed. 2d 246, 81 S. Ct. 1052 (1961).
Misunderstanding or lack of knowledge of the law or facts militates against a finding of wilfulness.
The taxpayer is less likely to be found willful if the tax return was prepared by a third party expert, such as a CPA.
 http://federaltaxcrimes.blogspot.ch/2014/06/comments-by-irs-personnel-on-new.html